Quantum Mechanics 3: the Quantum Mechanics of Many-Particle Systems

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Quantum Mechanics 3: the quantum mechanics of many-particle systems

W.J.P. Beenakker

Academic year 2017–2018

Contents of the common part of the course:

1) Occupation-number representation

2) Quantum statistics (up to 2.5.3) § Module 1 (high-energy physics):

2) Quantum statistics (rest)

3) Relativistic 1-particle quantum mechanics

4) Quantization of the electromagnetic ﬁeld

5) Many-particle interpretation of the relativistic quantum mechanics

The following books have been used:

F. Schwabl, “Advanced Quantum Mechanics”, third edition (Springer, 2005);

David J. Griﬃths, “Introduction to Quantum Mechanics”, second edition (Prentice Hall, Pearson Education Ltd, 2005);

Eugen Merzbacher, “Quantum Mechanics”, third edition (John Wiley & Sons, 2003);

B.H. Bransden and C.J. Joachain, “Quantum Mechanics”, second edition (Prentice Hall, Pearson Education Ltd, 2000). Contents

1 Occupation-number representation 1 1.1 Summary on identical particles in quantum mechanics ...... 1 1.2 Occupation-numberrepresentation ...... 5 1.2.1 ConstructionofFockspace...... 5 1.3 Switching to a continuous 1-particle representation ...... 12 1.3.1 Position and momentum representation ...... 14 1.4 Additive many-particle quantities and particle conservation ...... 15 1.4.1 Additive 1-particle quantities ...... 16 1.4.2 Additive 2-particle quantities ...... 19 1.5 Heisenberg picture and second quantization ...... 20 1.6 Examples and applications: bosonic systems ...... 23 1.6.1 The linear harmonic oscillator as identical-particle system . . . . . 23 1.6.2 Forced oscillators: coherent states and quasi particles ...... 26 1.6.3 Superfluidity ...... 31 1.6.4 Superfluidity for weakly repulsive spin-0 bosons (part 1) ...... 33 1.6.5 Intermezzo: the Bogolyubov transformation for bosons ...... 38 1.6.6 Superfluidity for weakly repulsive spin-0 bosons (part 2) ...... 40 1.6.7 The wonderful world of superfluid 4He: the two-fluid model . . . . 42 1.7 Examples and applications: fermionic systems ...... 44 1.7.1 Fermiseaandholetheory ...... 44 1.7.2 The Bogolyubov transformation for fermions ...... 46

2 Quantum statistics 49 2.1 Thedensityoperator(J.vonNeumann,1927) ...... 52 2.2 Example: polarization of a spin-1/2 ensemble ...... 55 2.3 Theequationofmotionforthedensity operator ...... 58 2.4 Quantum mechanical ensembles in thermal equilibrium ...... 59 2.4.1 Thermal equilibrium (thermodynamic postulate) ...... 61 2.4.2 Canonical ensembles (J.W. Gibbs, 1902) ...... 61 2.4.3 Microcanonicalensembles ...... 65 2.4.4 Grand canonical ensembles (J.W. Gibbs, 1902) ...... 66 2.4.5 Summary ...... 69 2.5 FermigasesatT=0 ...... 69 2.5.1 GroundstateofaFermigas ...... 74 2.5.2 Fermi gas with periodic boundary conditions ...... 76 2.5.3 Fermi-gas model for conduction electrons in a metal ...... 77 2.5.4 Fermi-gasmodelforheavynuclei ...... 80 2.5.5 Stars in the ﬁnal stage of stellar evolution ...... 84 2.6 Systems consisting of non-interacting particles ...... 87 2.7 Idealgasesinthreedimensions ...... 90 2.7.1 Classicalgases...... 92 2.7.2 Bose gases and Bose–Einstein condensation (1925) ...... 93 2.7.3 Fermi gases at T =0(noexammaterial)...... 98 6 2.7.4 Experimental realization of Bose–Einstein condensates ...... 101

3 Relativistic 1-particle quantum mechanics 103 3.1 First attempt: the Klein–Gordon equation (1926) ...... 104 3.2 Secondattempt: theDiracequation(1928) ...... 109 3.2.1 The probability interpretation of the Dirac equation ...... 111 3.2.2 Covariant formulation of the Dirac equation ...... 112 3.2.3 Dirac spinors and Poincar´etransformations ...... 113 3.2.4 Solutions to the Dirac equation ...... 116 3.2.5 ThesuccessstoryoftheDiractheory ...... 118 3.2.6 Problems with the 1-particle Dirac theory ...... 122

4 Quantization of the electromagnetic field 123 4.1 Classical electromagnetic fields: covariant formulation ...... 123 4.1.1 Solutions to the free electromagnetic wave equation ...... 125 4.1.2 Electromagnetic waves in terms of harmonic oscillations ...... 127 4.2 Quantizationandphotons ...... 129 4.2.1 Particle interpretation belonging to the quantized electromagnetic field130 4.2.2 Photon states and density of states for photons ...... 131 4.2.3 Zero-point energy and the Casimir effect ...... 132 4.2.4 Continuum limit ...... 133 4.3 A new perspective on particle –wave duality ...... 134 4.4 Interactions with non-relativistic quantum systems ...... 135 4.4.1 Classical limit ...... 138 4.4.2 Spontaneous photon emission (no exam material) ...... 139 4.4.3 The photon gas: quantum statistics forphotons ...... 141

5 Many-particle interpretation of the relativistic QM 144 5.1 Quantization of the Klein–Gordon ﬁeld ...... 145 5.2 Quantization of the Dirac ﬁeld ...... 149 5.2.1 Extra: the 1-(quasi)particle approximation for electrons ...... 153

A Fourier series and Fourier integrals i A.1 Fourierseries ...... i A.2 Fourier integrals ...... ii A.2.1 Deﬁnition of the δ function ...... iii

B Properties of the Pauli spin matrices iv

C Lagrange-multiplier method v

D Special relativity: conventions and deﬁnitions v

E Electromagnetic orthonormality relations ix 1 Occupation-number representation

In this chapter the quantum mechanics of identical-particle systems will be worked out in detail. The corresponding space of quantum states will be constructed in the occupation-number representation by employing creation and annihilation operators. This will involve the introduction of the notion of quasi particles and the concept of second quantization. Similar material can be found in Schwabl (Ch.1,2 and 3) and Merzbacher (Ch.21,22 and the oscillator part of Ch.14).

1.1 Summary on identical particles in quantum mechanics

Particles are called identical if they cannot be distinguished by means of specific intrinsic properties (such as spin, charge, mass, ). ··· This indistinguishability has important quantum mechanical implications in situations where the wave functions of the identical particles overlap, causing the particles to be observable simultaneously in the same spatial region. Examples are the interaction region of a scattering experiment or a gas container. If the particles are effectively localized, such as metal ions in a solid piece of metal, then the identity of the particles will not play a role. In those situations the particles are effectively distinguishable by means of their spatial coordinates and their wave functions have a negligible overlap.

For systems consisting of identical particles two additional constraints have to be imposed while setting up quantum mechanics (QM).

Exchanging the particles of a system of identical particles should have no observ- • able eﬀect, otherwise the particles would actually be distinguishable. This gives rise to the concept of permutation degeneracy, i.e. for such a system the expectation value for an arbitrary many-particle observable should not change upon interchanging the identical particles in the state function. As a consequence, quantum mechanical observables for identical-particle systems should be symmetric functions of the separate 1-particle observables.

Due to the permutation degeneracy, it seems to be impossible to ﬁx the quantum • state of an identical-particle system by means of a complete measurement. Nature has bypassed this quantum mechanical obstruction through the symmetrization postulate: identical-particle systems can be described by means of either totally symmetric state functions if the particles are bosons or totally antisymmetric state functions if the particles are fermions. In the non-relativistic QM it is an empirical fact that no mixed symmetry occurs in nature.

1 In this chapter the symmetrization postulate will be reformulated in an alternative way, giving rise to the conclusion that only totally symmetric/antisymmetric state functions will ﬁt the bill.

Totally symmetric state functions can be represented in the so-called q-representation by ψ (q , , q , t), where q , , q are the “coordinates” of the N separate identical par- S 1 ··· N 1 ··· N ticles. These coordinates are the eigenvalues belonging to a complete set of commuting 1-particle observablesq ˆ. Obviously there are many ways to choose these coordinates. A popular choice is for instance qj = (spatial coordinate ~rj , magnetic spin quantum number m σ , ), where the dots represent other possible internal (intrinsic) degrees of free- sj ≡ j ··· dom of particle j . For a symmetric state function we have

ˆ P ψS(q1, , qN , t) = ψS (qP (1) , , qP (N) , t) = ψS(q1, , qN , t) , (1) P∀ˆ ··· ··· ···

with Pˆ a permutation operator that permutes the sets of coordinates of the identical particles according to

q q , q q , , q q . (2) 1 → P (1) 2 → P (2) ··· N → P (N) Particles whose quantum states are described by totally symmetric state functions are called bosons. They have the following properties:

- bosons have integer spin (see Ch. 4 and 5);

- bosons obey so-called Bose–Einstein statistics (see Ch. 2);

- bosons prefer to be in the same quantum state.

Totally antisymmetric state functions can be represented in the q-representation by ψ (q , , q , t), with A 1 ··· N ˆ + ψA(q1, , qN , t) even P ˆ ··· P ψA(q1, , qN , t) = ψA(qP (1) , , qP (N) , t) = . (3) ∀ˆ ··· ···  P ψ (q , , q , t) odd Pˆ  − A 1 ··· N  A permutation Pˆ is called even/odd if it consists of an even/odd number of two-particle interchanges. Particles whose quantum states are described by totally antisymmetric state functions are called fermions. They have the following properties:

- fermions have half integer spin (see Ch. 5);

- fermions obey so-called Fermi–Dirac statistics (see Ch. 2);

- fermions are not allowed to be in the same quantum state.

2 If the above (anti)symmetrization procedure results in spatial symmetrization, the particles have an increased probability to be in each other’s vicinity. Note that this can apply to identical fermions if they happen to be in an antisymmtric spin state. On the other hand, spatial antisymmetrization gives rise to a decreased probability for the particles to be in each other’s vicinity. Note that this can apply to identical spin-1 bosons if they happen to be in an antisymmtric spin state.

Isolated non-interacting many-particle systems: many-particle systems with negi- gible interactions among the particles are called non-interacting many-particle systems. The properties of such systems are determined by the type(s) of particles involved and the possible influence of external potentials on the system (e.g. caused by a magnetic field). We speak of an ideal gas if the number of non-interacting particles is very large and if we place the system in a macroscopic finite enclosure.

Isolated non-interacting many-particle systems (such as free-particle systems, ideal gasses, ... ) play a central role in this lecture course, both for determining the physical properties of identical-boson/fermion systems and for setting up relativistic QM.

Since we are dealing here with isolated systems, the total energy is conserved and the state function can be expressed in terms of stationary states

ψ (q , , q , t) = exp( iEt/~) ψ (q , , q ) , E 1 ··· N − E 1 ··· N with Hˆ ψ (q , , q ) = E ψ (q , , q ) . (4) E 1 ··· N E 1 ··· N The Hamilton operator describing a non-interacting many-particle system comprises of pure 1-particle Hamilton operators, i.e. Hamilton operators that depend exclusively on observables belonging to individual single particles. Denoting the 1-particle Hamilton

operator of particle j by Hˆj , the N-particle Hamilton operator reads

N Hˆ = Hˆ , with Hˆ , Hˆ = 0 for all j,k =1, , N . (5) j j k ··· j =1 X This implies that the Hamilton operators of the individual particles are compatible observables, as expected for non-interacting particles. Suppose now that the 1-particle energy- eigenvalue equation ˆ Hj ψλj (qj) = Eλj ψλj (qj) (6) gives rise to an orthonormal set ψ (q ) of energy eigenfunctions belonging to the energy { λj j } eigenvalues Eλj , which are labeled by a complete set of quantum numbers. Examples of

such complete sets of quantum numbers are for instance λj = nj for a linear harmonic

3 oscillator or λj = (nj ,ℓj , mℓj , msj ) for a 1-electron atom. For the orthonormal set of energy eigenstates of the complete non-interacting N-particle system we can identify three scenarios.

A) The particles are distinguishable. The orthonormal set of N-particle energy eigenstates comprises of states of the form

N ψ (q , , q ) = ψ (q ) ψ (q ) ψ (q ) , with E = E . (7) E 1 ··· N λ1 1 λ2 2 ··· λN N λj j =1 X Such so-called product functions describe an uncorrelated system for which the properties of a speciﬁc particle can be measured without being inﬂuenced by the other particles.

The complete set of all product functions spans the space of all possible N-particle states involving distinguishable particles.

B) The particles are indistinguishable bosons. The orthonormal set of N-particle energy eigenstates comprises of totally symmetric states of the form 1 ψS (q1, , qN ) = ψλ1 (qP (1) ) ψλ2 (qP (2) ) ψλN (qP (N) ) , ··· S ··· N diﬀ.perm.X with = number of diﬀerent permutations of λ , ,λ . (8) NS 1 ··· N The possible energy eigenvaluesp are the same as in equation (7).

The space of bosonic N-particle states is spanned by a reduced set of linear combinations of product functions. As expected, this describes a correlated system for which the measurement of the properties of a speciﬁc particle is inﬂuenced by the other particles.

C) The particles are indistinguishable fermions. The orthonormal set of N-particle energy eigenstates comprises of totally antisymmetric states of the form 1 ψ (q , , q ) = ( 1)P ψ (q ) ψ (q ) ψ (q ) A 1 ··· N √ − λ1 P (1) λ2 P (2) ··· λN P (N) N! perm. X ψ (q ) ψ (q ) ψ (q ) λ1 1 λ2 1 ··· λN 1 ψλ (q2) ψλ (q2) ψλ (q2) 1 1 2 ··· N = . . . . , (9) √N! . . .. .

ψλ (qN ) ψλ (qN ) ψλ (qN ) 1 2 ··· N

where the determinant is known as the Slater determinant. The possible energy eigenvalues are again the same as in equation (7), bearing in mind that λ , ,λ 1 ··· N 4 have to be all diﬀerent. If two complete sets of quantum numbers λj and λk coincide, then two of the columns in the Slater determinant are identical and the totally

antisymmetric eigenfunction ψA vanishes. This is known as the

Pauli exclusion principle for identical fermions: no two identical fermions can be in the same fully speciﬁed 1-particle quantum state.

Also the space of fermionic N-particle states is spanned by a reduced set of linear combinations of product functions. This too describes a correlated system for which the measurement of the properties of a speciﬁc particle is inﬂuenced by the other particles.

1.2 Occupation-number representation

Our aim: we want to construct the space of quantum states (Fock space) for a many-particle system consisting of an arbitrary number of unspecified identical particles. This Fock space makes no statements about the physical scenario that the considered particles are in, such as being subject to interactions, external influences, etc.. It simply is the complex vector space (Hilbert space, to be more precise) that includes all possible many-particle states, on which the quantum mechanical many-particle theory should be formulated. The actual construction of Fock space involves finding a complete set of basis states for the decomposition of arbitrary many-particle state functions. The properties of these basis states will fix the properties of the Fock space.

The general rules for the construction of Fock space are:

states should not change when interchanging particles; • in order to guarantee the superposition principle, Fock space should not change when • changing the representation of the basis states.

1.2.1 Construction of Fock space

Consider identical particles of an unspeciﬁed type and assumeq ˆ to be a corresponding complete set of commuting 1-particle observables. Take the fully speciﬁed eigenvalues of these observables to be exclusively discrete: q labeled by j =1, 2, . The correspond- j ··· ing 1-particle basis of normalized eigenstates ofq ˆ is indicated by q , j = 1, 2, . {| ji ···} Subsequently we span the space of many-particle states by means of (special) linear combinations of product functions constructed from these 1-particle basis states, in analogy to the 1-particle energy eigenfunctions that were used in 1.1 to span the space of non- § interacting many-particle states. Since we are dealing with identical particles, a legitimate

5 many-particle state can make no statement about the identity of a particle in a speciﬁc 1-particle eigenstate. Such a many-particle state can at best make statements about the

number of particles nj that reside in a given fully speciﬁed 1-particle eigenstate belong-

ing to the eigenvalue qj . These numbers are called occupation numbers and can take the values 0, 1, (if allowed). ··· Postulate (replacing the symmetrization postulate): the collective set of hermitian number operators nˆ , nˆ , , which count the number of identical particles in each of the 1 2 ··· 1-particle quantum states q , q , , form a complete set of commuting many-particle | 1i | 2i ··· observables. The employed complete set of 1-particle observables qˆ can be chosen freely.

The hidden postulate aspect is that the state space for interacting particles can be constructed from non-interacting building blocks that are based on 1-particle observables. Since the complete set of 1-particle observables can be chosen freely, the superposition principle is automatically incorporated without any restrictions. This will guarantee that no mixed symmetry will occur in Fock space.

Hence, the corresponding set of normalized eigenstates n , n , will span Fock space | 1 2 ···i completely:

0-particle state : Ψ(0) 0, 0, vacuum state , | i ≡ | ···i ≡

1-particle states : Ψ(1) 0, , 0, n =1, 0, q , (10) | j i ≡ | ··· j ···i ≡ | ji ...

with nˆ , n , = n , n , , (11) j |··· j ···i j |··· j ···i where indicates the other occupation numbers. The representation of states corre- ··· sponding to this type of basis is called the occupation-number representation.

We will build up Fock space from the vacuum by adding particles to the basis states step

by step. To this end we introduce the creation operator aˆj† , which adds a particle with quantum number q to a basis state , n , according to j |··· j ···i

aˆ† , n , = n +1 exp iα ( , n , ) , n +1, . (12) j |··· j ···i j j ··· j ··· |··· j ···i p In principle the phase factor exp iα ( , n , ) could depend on the chosen basis state j ··· j ··· , n , as well as the label j . From the orthonormality of the many-particle basis it |··· j ···i

6 follows automatically that

(12) , nj, aˆ† , n′ , ==== n′ +1 exp iαj( , n′ , ) δn ,n′ +1 h··· ···| j |··· j ···i j ··· j ··· j j q deﬁnition ==== , n′ , aˆ , n , ∗ , h··· j ···| j |··· j ···i such that

aˆj , nj, = √nj exp iαj( , nj 1, ) , nj 1, (13) |··· ···i − ··· − ··· |··· − ···i holds for the annihilation operator aˆj . This deﬁnition of the creation and annihilation operators is consistent with the notion of the vacuum as a state with no particles, since

(0) (10),(13) (0) aˆ Ψ ==== 0 Ψ aˆ† = 0 . (14) j | i ⇒ h | j

Note thata ˆ† =â , which implies that the creation and annihilation operators themselves j 6 j are not observables. At this point we can fix the first part of the phase convention of (0) the many-particle basis by specifying the action ofa ˆ† anda ˆ on the basis states Ψ j j | i respectively Ψ(1) : | i (0) (1) (1) (0) aˆ† Ψ Ψ , aˆ Ψ δ Ψ exp iα (0, 0, ) = 1 . (15) j | i ≡ | j i j | k i ≡ jk | i ⇒ j ··· For arbitrary basis states we then have

(11),(12) nˆj, aˆk†=j , nj, , nk, ==== nj nj aˆk†=j , nj, , nk, = 0 , 6 |··· ··· ···i − 6 |··· ··· ···i

(11),(12) nˆ , aˆ† , n , ==== [n +1] n aˆ† , n , =a ˆ† , n , , j j |··· j ···i j − j j |··· j ···i j |··· j ···i resulting in the commutator relations

herm. conj. nˆ , aˆ† = δ aˆ† ======nˆ , aˆ = δ aˆ . (16) j k jk k ⇒ j k − jk k The above-deﬁned creation and annihilation operators can be linked directly to the associated number operators. To this end we consider the following matrix elements for arbitrary basis states: (13) , nj, aˆ† aˆj , n′ , ==== nj δn ,n′ . h··· ···| j |··· j ···i j j This matrix is duly diagonal and has the correct occupation numbers as eigenvalues, i.e.

nˆj =a ˆj† aˆj . (17)

7 Switching to another discrete 1-particle representation. Next we switch from the 1-particle basis q , j = 1, 2, belonging to the complete {| ji ···} set of observablesq ˆ to a 1-particle basis p , r =1, 2, belonging to the alternative {| ri ···} complete set of observablesp ˆ. From the completeness relations it follows that

Nˆ = nˆj . (22) j P This operator counts the total number of particles of a given identical-particle system, which should be invariant under an arbitrary basis transformation. Indeed we ﬁnd

(17) (21) (19) Nˆ ==== aˆ† aˆ ==== ˆb† ˆb p q q p ==== ˆb† ˆb δ = ˆb† ˆb . j j r v h r| jih j| vi r v rv r r j j,r,v r,v r X X X X

8 Construction of Fock space: part one.

For an arbitrary many-particle state Ψ it should hold thata ˆ† aˆ† Ψ anda ˆ† aˆ† Ψ fun- | i j k | i k j | i damentally describe the same state. This implies a relation of the type

aˆ† aˆ† exp iβ(Ψ,j,k) aˆ† aˆ† Ψ = 0 , j k − k j | i where the phase factor exp iβ(Ψ,j,k) can depend on the many-particle state Ψ as well | i as the labels j and k. Next we invoke the superposition principle to prove that exp iβ(Ψ,j,k) has to be independent of the many-particle state Ψ . Consider to this end two diﬀerent arbitrary | i many-particle basis states Ψ and Ψ in the q-representation. The superposition | 1i | 2i principle then states that also Ψ = c Ψ + c Ψ with c C should describe a | i 1 | 1i 2 | 2i 1,2 ∈ legitimate many-particle state. From this it follows that

c exp iβ(Ψ,j,k) aˆ† aˆ† Ψ + c exp iβ(Ψ,j,k) aˆ† aˆ† Ψ 1 k j | 1i 2 k j | 2i = exp iβ(Ψ,j,k) aˆ† aˆ† Ψ =a ˆ† aˆ† Ψ =a ˆ† aˆ† (c Ψ + c Ψ ) k j | i j k | i j k 1 | 1i 2 | 2i = c exp iβ(Ψ ,j,k) aˆ† aˆ† Ψ + c exp iβ(Ψ ,j,k) aˆ† aˆ† Ψ . 1 1 k j | 1i 2 2 k j | 2i Herea ˆ† aˆ† Ψ anda ˆ† aˆ† Ψ are again two diﬀerent basis states. Therefore k j | 1i k j | 2i

exp iβ(Ψ1,j,k) = exp iβ(Ψ2,j,k) exp iβ(j, k) , Ψ1∀,Ψ2 ≡ which implies that

aˆj† aˆk† exp iβ(j, k) aˆk† aˆj† Ψ = 0 aˆj† aˆk† exp iβ(j, k) aˆk† aˆj† = 0 . Ψ∀ − | i ⇒ − Subsequently the representation independence of Fock space can be used by rewriting this operator identity according to equation (21) in terms of an arbitrary alternative 1-particle representation, for which a similar operator identity holds:

p q p q ˆb† ˆb† exp iβ(j, k) ˆb† ˆb† = 0 and ˆb† ˆb† exp iβ′(r, v) ˆb† ˆb† = 0 . h r| jih v| ki r v − v r r v − v r r,v X For each unique (r, v)-combination in the p-representation we ﬁnd

p q p q exp iβ′(r, v) exp iβ(j, k) h r| jih v| ki − + p q p q 1 exp iβ(j, k) exp iβ′(r, v) = 0 . h v| jih r| ki − Since this has to hold true for an arbitrary basis transformation, we eventually obtain

exp iβ(j, k) = exp iβ′(r, v) = 1 exp(iβ) . ± ≡ 9 In this way we have found the phase factor exp(iβ) to be universal and to give rise to two branches of solutions:

aˆj†, aˆk† Ψ = 0 commutation relations : aˆj†, aˆk† = aˆj, aˆk = 0 Ψ∀ | i ⇒ or (23)

aˆj†, aˆk† Ψ = 0 anticommutation relations : aˆj†, aˆk† = aˆj, aˆk = 0 , Ψ∀ | i ⇒ introducing the anticommutator

A,ˆ Bˆ AˆBˆ + BˆA.ˆ (24) ≡ Construction of Fock space: part two. Finally this result can be combined with the general commutation identities (16). To this end we use the fact that a commutator of the type AˆB,ˆ Cˆ can be written both in terms of commutators and anticommutators:

AˆB,ˆ Cˆ = Aˆ B,ˆ Cˆ + A,ˆ Cˆ Bˆ = Aˆ B,ˆ Cˆ A,ˆ Cˆ Bˆ . (25) − In this way we ﬁnd for the commuting set of solutions in equation (23) that

(17) (23) (16) nˆj, ˆak† ==== aˆj† aˆj, aˆk† =a ˆj† aˆj, aˆk† + aˆj†, aˆk† aˆj ====a ˆj† aˆj, aˆk† ==== δjk aˆj† ,

and for the anticommuting set of solutions that

(17) (23) (16) nˆ , aˆ† ==== aˆ† aˆ , aˆ† =a ˆ† aˆ , aˆ† aˆ†, aˆ† aˆ ====a ˆ† aˆ , aˆ† ==== δ aˆ† . j k j j k j j k − j k j j j k jk j All in all the creation and annihilation operators should thus satisfy either

ˆ commutation relations : aˆj†, aˆk† = aˆj, aˆk = 0 , aˆj, aˆk† = δjk 1 , (26) the corresponding identical particles are called bosons,

ˆ anticommutation relations : aˆj†, aˆk† = aˆj, aˆk = 0 , aˆj, aˆk† = δjk 1 , (27) the corresponding identical particles are called fermions.

In the next chapter it will be shown that these two branches of creation and annihilation operators for identical many-particle systems give rise to utterly diﬀerent statistical theories. In the exercise class it will be proven that the (anti)commutation relations retain their form under an arbitrary change of representation, which implies that to a given type

10 of particle belongs a given type of statistics and that mixed statistics is indeed excluded.

In addition it will be shown that the many-particle states

n n nj (â†) 1 (â† ) 2 (â†) n , n , 1 2 Ψ(0) j Ψ(0) (28) | 1 2 ···i ≡ √ √ ···| i ≡ | i n1! n2! j nj! Y p form the orthonormal basis of Fock space that we set out to construct. This universal basis applies to bosons and fermions alike. Note, though, that the order of the creation operators matters in the fermionic case and does not matter in the bosonic case!

Properties of bosonic many-particle systems (see exercise 1):

The basis given above corresponds to the explicit phase convention • exp iα ( , n , ) 1 (29) j ··· j ··· ≡ in equation (12), such that

aˆj , nj, = √nj , nj 1, , |··· ···i |··· − ···i (30) aˆ† , n , = n +1 , n +1, . j |··· j ···i j |··· j ···i p For k-fold annihilation (k =0, 1, ) this becomes • ··· (ˆa )k , n , = n (n 1) (n k+1) , n k, , j |··· j ···i j j − ··· j − |··· j − ···i q which only leads to a sensible result if the series terminates as soon as n k < 0. j − From this we can deduce straightaway that nj can exclusively take on the values

n = 0, 1, 2, , (31) j ··· as expected for the eigenvalues of a number operator.

The order in which the creation operators occur in the basis states is irrelevant. As • a result, the state functions will be totally symmetric in this version of Fock space, as expected for bosonic states. This aspect will be worked out in 1.3. §

Properties of fermionic many-particle systems (see exercise 2):

2 1 As a result of equation (27) we know that (ˆa†) = aˆ†, aˆ† = 0. This we recognize • j 2 j j as the anticipated Pauli exclusion principle: no two fermions can occupy the same fully speciﬁed 1-particle state.

11 The occupation numbers can take on only two values, •

nj = 0, 1 . (32)

This follows from the fact that the number operators are projection operators in this case: aˆ2 =0 2 (17) (27) j nˆ ====a ˆ† aˆ aˆ† aˆ ====a ˆ† (1ˆ aˆ† aˆ )ˆa ====a ˆ† aˆ =n ˆ , j j j j j j − j j j j j j concurring with Pauli’s exclusion principle.

The basis given above corresponds to the explicit phase convention • j 1 − exp iα ( , n , ) ( 1)N

aˆ , n , = δ ( 1)N

This time the order in which the creation operators occur in the basis states does • matter and the state functions will be totally antisymmetric in this version of Fock space, as expected for fermionic states. This latter aspect will be worked out in 1.3 § The state functions of arbitrary identical-particle systems can now be expressed in terms of the basis states (28). Usually these state functions will describe a system with a ﬁxed number of particles. However, in particular in relativistic quantum mechanical situations

this may not be true, for instance when particles decay (such as n p e− ν¯ ) or when a → e photon is emitted during de-excitation of an excited atom (see Ch. 4). Maser and laser systems fall into this category as well, as in that case the radiation is built coherently (i.e. in phase). An example of such a coherent state will be worked out in 1.6. Finally, § as artefacts of certain approximation methods situations may arise in which the number of particles is eﬀectively not conserved anymore (see 1.6 and 1.7). § §

1.3 Switching to a continuous 1-particle representation

Since Fock space does not depend on the chosen 1-particle representation, we could also have opted for a complete set of commuting 1-particle observables kˆ with a continuous eigenvalue spectrum k . Examples are the position and momentum operators. The cor- { } responding “orthonormal” 1-particle basis is given by k . In this case it makes no {| i} sense to use an occupation-number representation based on counting, unless the continuous k-space is partitioned in small cells.

12 In analogy with the previous discussion we will introduce creation and annihilation operators

aˆ† aˆ†(k) anda ˆ aˆ(k) , (35) k ≡ k ≡ with the usual properties

(0) (0) aˆ†(k) Ψ k anda ˆ(k) Ψ = 0 . (36) | i≡| i | i The properties of these creation and annihilation operators follow directly from the disrete case by performing a 1-particle basis transformation

herm. conj. aˆ† = dk aˆ†(k) k q ======aˆ = dk aˆ(k) q k , j h | ji ⇒ j h j| i Z Z herm. conj. aˆ†(k) = aˆ† q k ======aˆ(k) = aˆ k q , (37) j h j| i ⇒ j h | ji j j X X which can be derived in analogy to equation (21) from

(0) compl. (36) (0) aˆ† Ψ = q ==== dk k k q ==== dk k q ˆa†(k) Ψ . j | i | ji | ih | ji h | ji | i Z Z For this basis transformation the following unitarity conditions hold:

Subsequently equations (26) and (27) for a discrete 1-particle representation can be used to obtain

aˆ†(k), aˆ†(k′) = aˆ(k), aˆ(k′) = 0 , aˆ(k), aˆ†(k′) = δ(k k′)1ˆ (39) − for a bosonic system in a continuous 1-particle representation and

aˆ†(k), aˆ†(k′) = aˆ(k), aˆ(k′) = 0 , aˆ(k), aˆ†(k′) = δ(k k′)1ˆ (40) − for a fermionic system in a continuous 1-particle representation. Proof: the derivation of the vanishing (anti)commutators is trivial. In addition

(37) (26),(27) aˆ(k)ˆa†(k′) aˆ†(k′)ˆa(k) ==== k q q ′ k′ aˆ aˆ†′ aˆ†′ aˆ ==== k q q k′ 1ˆ ∓ h | jih j | i j j ∓ j j h | jih j| i j,j′ j X X (38) ==== δ(k k′)1ˆ , − 13 where the minus (plus) sign refers to bosonic (fermionic) systems.

The representation-independent total number operator Nˆ can in this case be expressed in terms of the number density operator

nˆ(k) aˆ†(k)ˆa(k) (41) ≡ according to

(22) (17) (37) Nˆ ==== nˆ ==== aˆ† aˆ ==== dk dk′ aˆ†(k)ˆa(k′) k q q k′ j j j h | jih j| i j j j X X Z Z X (38) (41) ==== dk dk′ aˆ†(k)ˆa(k′)δ(k k′) ==== dk nˆ(k) . (42) − Z Z Z The basis of N-particle state functions takes the following form in the k-representation:

1 (0) k1, ,kN aˆ†(k1) aˆ†(kN ) Ψ : k1 , ,kN k . (43) | ··· i ≡ √N! ··· | i ··· ∈{ } n o This is a direct consequence of the fact that

dk dk k , ,k k , ,k = 1ˆ (44) 1 ··· N | 1 ··· N ih 1 ··· N | N Z Z is the unit operator in the N-particle subspace, which will be proven in exercise 3. An arbitrary N-particle state function Ψ can now be represented in the k-representation by | i the function ψ(k , ,k ) k , ,k Ψ . (45) 1 ··· N ≡ h 1 ··· N | i As expected, this function is totally symmetric for bosonic systems and totally antisymmetric for fermionic systems. This too will be proven in exercise 3.

1.3.1 Position and momentum representation

The previous discussion can be generalized trivially to cases with mixed 1-particle eigenvalue spectra, which are partially discrete and partially continuous. Two examples of this will be worked out now.

The position representation: as a ﬁrst example of a mixed representation we consider spin-s particles in the position representation. This representation will be used in chapter 5 when setting up relativistic QM. In the previous expressions we now simply have to substitute

kˆ (~r,ˆ Sˆ /~) , k ~r IR3 , σ = m = s, s+1, ,s 1,s , → z { } → { ∈ s − − ··· − }

dk d~r , δ(k k′) δ ′ δ(~r ~r ′) anda ˆ(k) ψˆ (~r ) . (46) → − → σσ − → σ σ Z X Z 14 Quite suggestively the annihilation oparatora ˆ(k) is replaced in the position representation by the so-called ﬁeld operator ψˆσ(~r ). This suggestive notation is introduced in order to facilitate the discussion in 1.5. § The momentum representation: as a second example of a mixed representation we consider spin-s particles in the momentum representation. This representation is particularly useful for systems that (in ﬁrst approximation) consist of free particles, as will be used in chapter 5 for analysing relativistic quantum mechanical wave equations. In this case we have to perform the following substitutions:

kˆ (~p,ˆ Sˆ /~) , k ~p IR3 , σ = m = s, s+1, ,s 1,s , → z { } → { ∈ s − − ··· − }

dk d~p , δ(k k′) δ ′ δ(~p ~p ′) anda ˆ(k) aˆ (~p ) . (47) → − → σσ − → σ σ Z X Z These two representations are connected through a Fourier transformation:

(37) exp(i~p ~r/~) ψˆ (~r ) ==== d~p ~r ~p aˆ (~p ) = d~p · aˆ (~p ) , σ h | i σ (2π~)3/2 σ Z Z

(37) exp( i~p ~r/~) aˆ (~p ) ==== d~r ~p ~r ψˆ (~r ) = d~r − · ψˆ (~r ) . (48) σ h | i σ (2π~)3/2 σ Z Z In fact this is identical to the connection between the state functions in position and momentum space known from 1-particle QM (see the lecture course Quantum Mechanics 2).

1.4 Additive many-particle quantities and particle conservation

Observables: we have succeeded in constructing the state space applicable to systems consisting of an arbitrary number of identical particles and we have investigated what the basis states look like if we employ discrete and/or continuous 1-particle representations. As a next step in the construction of the many- particle toolbox we need to have a closer look at the many-particle observables.

In Fock space various kinds of many-particle observables are possible, as long as they are duly symmetric under particle interchange. An important class of many-particle observables consists of observables that belong to so-called additive many-particle quantities. These quantities can be categorized by the number of particles that participate as a clus- ter in the physical quantity. Subsequently these clusters will be summed/integrated over. A characteristic property of these additive many-particle quantities is the existence of a speciﬁc representation that makes the corresponding many-particle observable expressible solely in terms of number (density) operators. This automatically implies that diﬀerent eigenstates of the total number operator Nˆ are not mixed by such a many-particle observable (which entails particle conservation).

15 Additive many-particle quantities play a central role in setting up (free-particle) relativistic QM and in the description of systems consisting of weakly interacting identical particles (such as electrons in an atom or conduction electrons in a metal). In lowest-order approximation the particles of a weakly interacting system can be considered non-interacting, which causes the total energy to be expressible as a sum of individual 1-particle energies (see 1.1). As a ﬁrst-order correction to this, the interactions between pairs of particles can § be added. In this way the particles contribute in sets of two to the energy correction. This procedure can be extended trivially to allow for (progressively smaller) energy corrections involving larger clusters of particles.

1.4.1 Additive 1-particle quantities

Consider a 1-particle observable Aˆ. By employing the completeness relation for the discrete q-representation, this observable can de written as

Aˆ = qj qj Aˆ qj′ qj′ . ′ | ih | | ih | Xj,j Each individual term occurring in this sum brings the particle from a state q ′ to a | j i state q , with the matrix element q Aˆ q ′ as corresponding weight factor. Using this | ji h j| | j i 1-particle observable a proper many-particle observable can be constructed:

ˆ(1) ˆ Atot = Aα , α X ˆ ˆ(1) where Aα is the 1-particle observable belonging to particle α. The action of Atot in Fock space is to simply annihilate a particle in the state q ′ and subsequently create a particle | j i in the state q with weight factor q Aˆ q ′ : | ji h j| | j i

(1) ˆ ˆ ′ ′ ˆ ′ ′ Atot = aˆj† qj A qj aˆj = qj A qj aˆj† aˆj . (49) ′ h | | i ′ h | | i Xj,j Xj,j We speak of an additive 1-particle quantity if the corresponding many-particle observable can be represented in the form (49). This deﬁnition hinges on the fact that the given expression has exactly the same form for any discrete 1-particle representation:

(1) (18),(21) compl. ˆ ˆ ˆ ′ ′ ′ ˆ ′ ˆ ˆ ′ ˆ ′ Atot ==== br† pr qj qj A qj qj pr br ==== br† pr A pr br . ′ ′ h | ih | | ih | i ′ h | | i r,rX,j,j Xr,r Many-particle observables belonging to additive 1-particle quantities can also be translated trivially to a continuous 1-particle representation. By means of the basis transformation (37) and the unitarity conditions (38), equation (49) changes into

(1) Aˆ = dk dk aˆ†(k ) k Aˆ k aˆ(k ) = dk dk k Aˆ k aˆ†(k )ˆa(k ) . (50) tot 1 2 1 h 1| | 2i 2 1 2 h 1| | 2i 1 2 Z Z Z Z

16 The general expressions (49) and (50) are valid irrespective of the 1-particle observable Aˆ having discrete or continuous eigenvalues.

Special cases: let’s assume that the 1-particle observable Aˆ has discrete eigenvalues and is part of the complete set of commuting 1-particle observablesq ˆ. In that case

(49) (1) q Aˆ q ′ A ′ q q ′ = A δ ′ === Aˆ = A nˆ . h j| | j i ≡ j h j| j i j jj ⇒ tot j j j X The special feature of this representation is the fact that it merely involves counting particles, in analogy to the total number operator Nˆ. We actually already exploited this concept in 1.1 while determining the total energy of a non-interacting many-particle sys- § tem. Apart from additive 1-particle quantities with discrete eigenvalues, also additive 1-particle quantities exist with continuous eigenvalues (see below). In that case a continuous representation can be found such that

(50) k Aˆ k A(k ) k k = A(k )δ(k k ) === Aˆ(1) = dk A(k )ˆn(k ) . h 1| | 2i ≡ 2 h 1| 2i 1 1 − 2 ⇒ tot 1 1 1 Z Particle conservation: a characteristic feature of additive many-particle quantities is that the observables only consist of terms with as many creation as annihilation operators. As such these observables commute with the total number operator Nˆ and therefore do not mix the diﬀerent eigenstates of Nˆ. In that respect one could speak of “conservation of the number of particles”.

Additive 1-particle quantities in position and momentum representation. In view of future applications, we now take a closer look at a few useful additive 1-particle quantities in the mixed representations belonging to spin-s particles in either position or momentum space.

The total momentum: in the momentum representation the 1-particle momentum operator ~pˆ is part of the complete set of observables with respect to which the many-particle Fock space is formulated. As a result, the total momentum operator takes on a particularly simple form in the momentum representation (i.e. merely involving number density operators in momentum space):

ˆ (1) ~ P~ = d~p~p nˆ (~p ) = d~r ψˆ† (~r ) i~ ψˆ (~r ) . (51) tot σ σ − ▽ σ σ σ X Z X Z The proof of the last step in this equation involves the Fourier transformation (48) for switching from momentum to position representation (see also App. A for additional details):

17 ~ ~ exp( i~p ~r/ ) ~ i~ d~r ψˆ† (~r ) ψˆ (~r ) = i~ d~p d~p ′ aˆ† (~p )ˆa (~p ′) d~r − · exp(i~p ′ ~r/~) − σ ▽ σ − σ σ (2π~)3 ▽ · Z Z Z Z

exp(i[~p ′ ~p ] ~r/~) = d~p d~p ′ ~p ′ aˆ† (~p )ˆa (~p ′) d~r − · σ σ (2π~)3 Z Z Z

= d~p d~p ′ ~p ′ aˆ† (~p )ˆa (~p ′) δ(~p ~p ′) = d~p~p nˆ (~p ) . σ σ − σ Z Z Z

The total kinetic energy: the corresponding 1-particle observable Tˆ = ~pˆ 2/(2m) is again diagonal in the momentum representation. In analogy to the previous example we ﬁnd for the total kinetic-energy operator:

2 ~2 2 (1) ~p ~ Tˆ = d~p nˆ (~p ) = d~r ψˆ† (~r ) ψˆ (~r ) . (52) tot 2m σ σ − 2m ▽ σ σ σ X Z X Z

The total spin in the z-direction: in this case the corresponding 1-particle observable Sˆz is part of the complete set of observables of both the position and momentum representation. The total spin operator in the z-direction trivially reads:

ˆ (1) S~ ~e = d~p ~σ nˆ (~p ) = d~r ~σ nˆ (~r ) ~σNˆ , (53) tot · z σ σ ≡ σ σ σ σ X Z X Z X

where Nˆσ counts the total number of particles with spin component ~σ along the z-axis.

The total potential energy in an external ﬁeld: assume the particles to be subjected to an external potential V (~r ). The corresponding 1-particle observable V (~rˆ) is obviously diagonal in the position representation, resulting in the following total potential-energy operator:

(1) Vˆ d~rV (~r )ˆn (~r ) = d~k d~p (~k)â† (~p )â (~p ~~k) , (54) tot ≡ σ V σ σ − σ σ X Z X Z Z using the definition V (~r ) d~k (~k) exp(i~k ~r ). The derivation of the right-hand side ≡ V · of this equation follows the lineR of proof worked out in exercise 4 for a related type of potential.

Second quantization: the additive many-particle observables discussed above resemble quantized versions of the corresponding expectation values of the 1-particle observables in ordinary 1-particle QM, i.e. expectation values with ψ ψˆ. This formal connection → between 1-particle and many-particle quantum theory is indicated somewhat suggestively by the term “second quantization”.

18 1.4.2 Additive 2-particle quantities

In the discrete q-representation the ordered 2-particle basis states are given by

(2) (0) Ψ 0, , 0, n =1, 0, , 0, n =1, 0, =a ˆ† aˆ† Ψ , | j,k>ji ≡ | ··· j ··· k ···i j k | i 2 † (2) (ˆaj) (0) Ψ 0, , 0, nj =2, 0, = Ψ , (55) | jj i ≡ | ··· ···i √2 | i where the last pair state only has relevance for bosonic systems. With the help of the corresponding completeness relation, a 2-particle observable Bˆ can be written as

ˆ (2) (2) ˆ (2) (2) B = Ψjk Ψjk B Ψj′k′ Ψj′k′ . ′ ′ | ih | | ih | j,jX,k,k (2) Each individual term occurring in this sum brings the particle pair from a state Ψj′k′ (2) (2) (2) | i to a state Ψ , with the matrix element Ψ Bˆ Ψ ′ ′ as corresponding weight factor. | jk i h jk | | j k i Using this 2-particle observable a proper many-particle observable can be constructed:

1 Bˆ(2) = Bˆ , tot 2 αβ α,β=α X6 where Bˆ is the 2-particle observable belonging to particles α and β = α. The factor αβ 6 of 1/2 is introduced here to avoid double counting. The corresponding action of this observable in Fock space simply reads

(2) 1 (2) (2) 1 (2) (2) ˆ ˆ ′ ′ ˆ ′ ′ Btot = aˆj† aˆk† Ψjk B Ψj′k′ aˆk aˆj = Ψjk B Ψj′k′ aˆj† aˆk† aˆk aˆj . (56) 2 ′ ′ h | | i 2 ′ ′ h | | i j,jX,k,k j,jX,k,k This method of writing all creation operators on the left and all annihilation operators on the right in a many-particle operator is usually referred to as normal ordering. We speak of an additive 2-particle quantity if the corresponding many-particle observable can be represented in the form (56). This type of expression has the same form for any discrete 1-particle representation, bearing in mind that just like we have seen in 1.4.1 § each creation/annihilation operator in the expression is linked to a corresponding annihilation/creation operator that is hidden in one of the basis states in the matrix element. Switching to a continuous representation yields accordingly

(2) 1 Bˆ = dk dk k ,k Bˆ k ,k aˆ†(k )â†(k )â(k )â(k ) . (57) tot 2 1 ··· 4 h 1 2| | 3 4i 1 2 4 3 Z Z

Again the generic expressions (56) and (57) are valid irrespective of the 2-particle observ- abele Bˆ having discrete or continuous eigenvalues.

19 Special cases: let’s assume that the 2-particle observable Bˆ has discrete eigenvalues and is diagonal in the q-representation. Then

(2) (2) (2) (2) Ψ Bˆ Ψ ′ ′ B ′ ′ Ψ Ψ ′ ′ = B δ ′ δ ′ h jk | | j k i ≡ j k h jk | j k i jk jj kk

(56) (2) 1 1 === Bˆ = B aˆ† aˆ† aˆ aˆ = B (ˆn nˆ nˆ δ ) , ⇒ tot 2 jk j k k j 2 jk j k − j jk Xj,k Xj,k where we have used that

(16),(17) (17) aˆ† aˆ† aˆ aˆ ====a ˆ† aˆ aˆ† aˆ aˆ† δ aˆ ====n ˆ nˆ nˆ δ = pair number operator . (58) j k k j j j k k − j jk j j k − j jk Of course, additive 2-particle quantites may also have continuous eigenvalues (see below). In that case a continuous representation exists such that Bˆ becomes diagonal, allowing to ˆ(2) express Btot in terms of pair density operators belonging to that representation.

Example: spatial pair interactions in position and momentum representation. Consider a system comprising of spin-s particles with spatial pair interactions described by the observable U(~rˆ ~rˆ ). This observable is obviously diagonal in the position repre- 1 − 2 sentation, resulting in the following total operator for spatial pair interactions:

(2) 1 Uˆ d~r d~r U(~r ~r ) ψˆ† (~r )ψˆ† (~r )ψˆ (~r )ψˆ (~r ) . (59) tot ≡ 2 1 2 1 − 2 σ1 1 σ2 2 σ2 2 σ1 1 σ ,σ X1 2 Z Z In the momentum representation this becomes (see exercise 4)

(2) 1 Uˆ = d~k d~p d~p (~k)â† (~p )â† (~p )â (~p + ~~k)â (~p ~~k) , (60) tot 2 1 2 U σ1 1 σ2 2 σ2 2 σ1 1 − σ ,σ X1 2 Z Z Z using the definition U(~r ) d~k (~k) exp(i~k ~r ). ≡ U · R 1.5 Heisenberg picture and second quantization

In preparation for the relativistic treatment of quantum mechanics in chapter 5, we now address the time dependence of the creation and annihilation operators in the Heisenberg picture of quantum mechanics.

Consider to this end an isolated identical-particle system described by a Hamilton operator of the following form: Hˆ = Hˆ (1) + C 1ˆ (C IR) , (61) tot tot ∈ where the term Hˆ (1) is an additive observable of the type described in 1.4.1. Next tot § we address the time evolution of this many-particle system in the Heisenberg picture. As explained in the lecture course Quantum Mechanics 2, all state functions are time- independent in the Heisenberg picture and all time dependence of the system resides in

20 the time evolution of the operators. The diﬀerences with the many-particle formulation in the Schr¨odinger picture, which we have used so far, can be summarized as follows:

Schr¨odinger picture : state functions Ψ(t) = exp iHˆ t/~ Ψ(0) , → | i − tot | i observables Aˆ(t) , → (62) Heisenberg picture : state functions Ψ (t) = Ψ(0) , → | H i | i observables Aˆ (t) = exp iHˆ t/~ Aˆ(t) exp iHˆ t/~ , → H tot − tot where we have assumed that both pictures coincide at the time stam p t = 0. For example we have Hˆ (t) = Hˆ , (63) totH tot since we are dealing here with an isolated system. In the Heisenberg picture the operators satisfy the so-called Heisenberg equation of motion ˆ d ˆ i ˆ ˆ ∂A(t) AH (t) = AH (t), Htot + . (64) dt − ~ ∂t H Next we investigate the implications for the creation and annihilation operators.

The discrete case: for a generic discrete 1-particle representation we get

(1) (49) (63) Hˆ ==== pr Hˆ pr′ ˆb† ˆbr′ ==== pr Hˆ pr′ ˆb† (t)ˆbr′ (t) , tot r rH H ′ h | | i ′ h | | i Xr,r Xr,r with Hˆ being the 1-particle Hamilton-operator belonging to the additive part of the many- particle operator Hˆtot . From this it follows that

(26),(27) ˆ ˆ ˆ ′ ˆ ˆ ˆ ′ ˆ ′ ˆ ′ bv (t), Htot = pr H pr bv (t), br† (t)br (t) ==== pr H pr br (t)δvr H h | | i H H H h | | i H r,r′ r,r′ X X

= p Hˆ p ′ ˆb ′ (t) . v r rH ′ h | | i Xr The Heisenberg equation of motion then reads

d i~ ˆbv (t) = pv Hˆ pr ˆbr (t) , (65) dt H h | | i H r X where we have used that the creation and annihilation operators are time-independent in the Schr¨odinger picture. This equation of motion strongly resembles the Schr¨odinger equation for a 1-particle state function written in the p-representation:

Free particles in position space: in the Schr¨odinger picture the position-space annihi-

lation operator is given by the time-independent ﬁeld operator ψˆσ(~r ). The corresponding time-dependent operator in the Heisenberg picture is concisely written as

ψˆσ(~r ) (t) ψˆσ(~r, t) , (66) H ≡ which can be translated to momentum space by means of the Fourier transformation

exp(i~p ~r/~) ψˆσ(~r, t) = d~p · aˆσ (~p, t) . (67) (2π~)3/2 H Z ˆ(1) For free particles the Hamilton operator is given by the total kinetic-energy operator Ttot : 2 2 (52) ~p ′ (63) ~p ′ Hˆ ==== d~p ′ nˆ ′ (~p ′) ==== d~p ′ nˆ ′ (~p ′, t) . (68) tot σ σH ′ 2m ′ 2m Xσ Z Xσ Z In the momentum representation the Heisenberg equation of motion reads

2 2 ∂ (64),(68) ~p ′ (16) ~p i~ aˆσ (~p, t) ======d~p ′ aˆσ (~p, t), nˆσ′ (~p ′, t) ==== aˆσ (~p, t) ∂t H 2m H H 2m H σ′ Z X ~p 2 aˆσ (~p, t) = exp( iE~p t/~)ˆaσ(~p ) with E~p . (69) ⇒ H − ≡ 2m ˆ This implies the following for the ﬁeld operator ψσ(~r, t) in the position representation: ~ 2 ∂ ˆ (67),(69) exp(i~p ~r/ ) ~p i~ ψσ(~r, t) ======d~p · aˆσ (~p, t) ∂t (2π~)3/2 2m H Z

2 (67) ∂ ~ 2 === i~ ψˆ (~r, t) = ~ ψˆ (~r, t) . (70) ⇒ ∂t σ − 2m ▽ σ

In the Heisenberg picture the many-particle field operator ψˆ (~r, t) satisfies an equa- • σ tion that is identical to the free Schrödinger equation for a 1-particle wave function 1 in position space. For this reason ψˆσ(~r, t) is also known as the Schrödinger field. Equation (70) can be regarded as a quantized version of the “classical” Schrödinger equation for quantum mechanical wave functions. Also this direct link between a 1-particle wave equation and a many-particle theory formulated in the Heisenberg picture is a manifestation of the concept of second quantization.

1This relation extends to situations where potential-energy terms are added to the Hamilton operator.

22 Second quantization will play a crucial role in the transition from non- relativitic to relativistic QM. We will be forced to abandon the idea of a 1-particle quantum mechanical theory based on a relativistic wave equation and replace it by a corresponding many-particle formulation that constitutes the starting point for quantum ﬁeld theory.

As we have seen, the above-given equation of motion (wave equation) for ψˆ (~r, t) • σ can be solved (decoupled) straightforwardly by means of a Fourier decomposition (plane-wave expansion):

(67),(69) exp(i~p ~r/~) ψˆ (~r, t) ======d~p · exp( iE t/~)ˆa (~p ) . (71) σ (2π~)3/2 − ~p σ Z In fact this is a manifestation of particle–wave duality, as each annihilation/creation of a particle is automatically accompanied by a corresponding plane-wave factor. At a later stage we will make use of this aspect to come up with an alternative, many-particle formulation of relativistic QM.

1.6 Examples and applications: bosonic systems

In the preceding discussion we have systematically used the generic term “particles” to talk about the fundamental quantum-mechanical objects of a speciﬁc quantum system. It seems obvious what these particles should be: intuitively we expect that the particles belonging to an atomic gas system should be the atoms themselves. However, in the remainder of this chapter we will see that the particle interpretation of quantum systems in terms of the corresponding fundamental energy quanta can change radically if the interaction parameters of the system change. This will teach us to be more ﬂexible with the term “particles” than we are used to. This is actually one of the most profound new aspects of many-particle quantum mechanics that is covered in this lecture course.

1.6.1 The linear harmonic oscillator as identical-particle system

We kick oﬀ with one of the simplest and at the same time one of the most widely used quantum systems: the linear harmonic oscillator. Why the oscillator has such an important role to play in quantum physics will become clear at a later stage.

A linear harmonic oscillator is by deﬁnition a 1-particle system. However, it also possesses the hallmark features of an identical-particle system, which will play a crucial role in the description of particles in the context of relativistic QM. A completely worked out example along this line will be the photon description of electromagnetic ﬁelds in chapter 4. In order

23 to highlight the analogy between a linear harmonic oscillator and an identical-particle system we ﬁrst rewrite the Hamilton operator:

pˆ2 1 ~ω Hˆ = x + mω2 xˆ2 Pˆ2 + Qˆ2 , 2m 2 ≡ 2 1 mω with Pˆ pˆ and Qˆ xˆ . (72) ≡ x m~ω ≡ ~ r r Subsequently we introduce the raising, lowering en number operator according to 1 raising operator :a ˆ† Qˆ iPˆ , ≡ √2 − 1 lowering operator :a ˆ Qˆ + iPˆ , ≡ √2 number operator :n ˆ aˆ† aˆ . (73) ≡ The dimensionless position and momentum operators Qˆ and Pˆ have the following fundamental properties:

aˆ +ˆa† aˆ aˆ† 1 Qˆ† = Qˆ = , Pˆ† = Pˆ = − and Q,ˆ Pˆ = x,ˆ pˆx = i1ˆ . (74) √2 i√2 ~ In the last step we have used that the coordinate x and momentum px form a conjugate pair, which implies that the corresponding operators satisfy the usual (canonical) quantum mechanical quantization conditions. From this we can deduce the following features:

aˆ† =a ˆ, i.e. raising and lowering operators are not hermitian and therefore do not • 6 correspond to an observable quantity.

nˆ† = (â† aˆ)† =n ˆ , i.e. the number operator does correspond to an observable quantity. • the raising and lowering operators satisfy bosonic commutation relations: • 1 (74) a,ˆ aˆ† = Qˆ + iP,ˆ Qˆ iPˆ ==== 1ˆ and aˆ†, aˆ† = a,ˆ aˆ = 0 . (75) 2 − from the relation • 2 2 2 2 (74) aˆ aˆ† aˆ +â† Pˆ + Qˆ ==== − + =a ˆ† aˆ +âaˆ† i√2 √2 it follows that the Hamilton operator is linked directly to the number operator: ~ ~ ω 2 2 ω (75) 1 Hˆ = Pˆ + Qˆ = aˆ† aˆ +âaˆ† ==== ~ω nˆ + 1ˆ . (76) 2 2 2

24 A linear harmonic oscillator eﬀectively behaves like an identical-particle system consisting of non-interacting bosons. These bosons can take on precisely one value for the 1-particle

energy, i.e. E1 = ~ω. As such they can be regarded as the energy quanta belonging to

the considered vibrational/oscillatory motions. These quanta are created bya ˆ†, annihi- 1 ~ lated bya ˆ and counted byn ˆ . Except for a constant term (zero-point energy) 2 ω, the Hamilton-operator takes the form of a many-particle observable belonging to an additive 1-particle quantity.

The fundamental energy quanta of an oscillator system can be interpreted as non-interacting bosonic particles. Nature displays many diﬀerent bosonic vibration/oscillation quanta, in particular those that provide a particle interpretation to wave phenomena in classical physics. Well-known examples are photons, related to electromagnetic waves (see Ch.4), and phonons, related to collective vibrational modes of lattices and other sound waves (see exercise 5).

In accordance with 1.2.1 the energy eigenstates of the linear harmonic oscillator can be § written as

n (28) (ˆa ) (11) n ==== † 0 , with Hˆ n ==== ~ω n+ 1 n (n =0, 1, 2, ) . (77) | i √n! | i | i 2 | i ··· The meaning of the designations “raising” and “lowering” becomes apparent if we consider the action of the creation and annihilation operators:

(30) 1 aˆ† n ==== √n +1 n+1 Hˆ aˆ† n = ~ω n +1+ aˆ† n , | i | i ⇒ | i 2 | i

(30) aˆ n ==== √n n 1 Hˆ aˆ n = ~ω n 1+ 1 aˆ n . (78) | i | − i ⇒ | i − 2 | i The raising (lowering) operator changes a given energy eigenstate into the energy eigenstate belonging to the next excited (previous de-excited) energy level.

Remark: if we would have considered a fermionic oscillator

Hˆ = ~ω nˆ + C 1ˆ = ~ω aˆ† aˆ + C 1ˆ (C IR) , ∈ with a,ˆ aˆ† = 1ˆ and aˆ†, aˆ† = a,ˆ aˆ = 0 , (79) Pauli’s exclusion principle would have allowed for only two energy eigenvalues, ~ωC and

~ω(C + 1) corresponding to n = 0 and n = 1 respectively. Still, the operatorsa ˆ† anda ˆ would have been raising and lowering operators, since the commutator identity a,ˆ nˆ =ˆa holds equally well in the fermionic case.

25 1.6.2 Forced oscillators: coherent states and quasi particles

Let’s now add an interaction to the linear harmonic oscillator of 1.6.1 according to § 1 1 Hˆ = ~ω nˆ λaˆ† λ∗ aˆ + 1ˆ = ~ω nˆ + 1ˆ ~ω(Reλ) aˆ† +ˆa ~ω(Imλ) iaˆ† iaˆ − − 2 2 − − − 2 (72),(74) pˆ 1 ==== x + mω2 xˆ2 xˆ Reλ√2m~ω3 pˆ Imλ 2~ω/m (λ C) . (80) 2m 2 − − x ∈ Systems of this type are called forced linear harmonicp oscillators. They are omnipresent in nature. For instance, taking λ = q /√2m~ω3 and IR we obtain an interaction E E ∈ term of the form Vˆ = qxˆ , which we recognize as the interaction term of a charged − E linear harmonic oscillator with charge q subjected to a classical electric ﬁeld along the E x-axis. The striking feature of the additional interaction term is that it does not conserve the number of oscillator quanta, since

(16) n,ˆ Hˆ = ~ω n,ˆ λaˆ† + λ∗ aˆ ==== ~ω(λaˆ† λ∗ aˆ) = 0 . − − − 6 This is a ﬁrst example of what is referred to as a non-additive interaction, i.e. an interaction that does not conserve the total number of particles.

A new particle interpretation: nature is riddled with systems that involve non-additive interactions. For instance, such systems are ubiquitous in particle physics (e.g. through the production and decay of particles), quantum optics/spectroscopy (e.g. in laser systems and when photons are being absorbed/emitted by atoms) and quantum electronics (e.g. in describing charge ﬂows).

In scenarios involving non-additive interactions it usually pays to switch to an alternative particle interpretation, i.e. an alternative description of the interacting quantum system in terms of non-interacting energy quanta that are called quasi particles. Expressed in terms of these new energy quanta the Hamilton operator will take on an additive 1-particle form, which causes the number of quasi particles to be conserved. Furthermore, this implies the existence of a representation that will make the Hamilton operator expressible in terms of number operators for quasi particles.

To get a feel for what a quasi particle could be, consider a particle that propagates through a medium while influencing other particles and while being influenced by other particles. The combination of particle and ambient influences can collectively behave as a separate free entity. For instance, a conduction electron propagating through a piece of metal can excite vibrations (phonons) in the metal-ion lattice, while in turn the lattice vibrations can affect the motions of the conduction electrons. The degree by which the quasi particles will differ from the original particles depends on the nature and strength of the ambient interactions.

26 In all examples covered in this lecture course, the original particle interpretation is retrieved in the limit of vanishing ambient inﬂuences. If the original particles are bosons (fermions) then the same should be true for the quasi particles.

Therefore, while switching from particles to quasi particles, one set of bosonic (fermionic) creation and annihilation operators will be replaced by another set. As such, it will not come as a surprise that a particle–quasi particle transition actually boils down to performing a unitary transformation in Fock space, since operator identities such as bosonic commutation relations (fermionic anticommutation relations) will be invariant in that case.

Quasi particles and coherent states (Roy J. Glauber, 1963): in case of a forced

linear harmonic oscillator we can switch to new bosonic quasi-particle operatorsc ˆ andc ˆ† by means of the transformation

cˆ =a ˆ λ1ˆ , cˆ† =a ˆ† λ∗ 1ˆ c,ˆ cˆ† = a,ˆ aˆ† = 1ˆ . (81) − − ⇒ The number operator for quasi particles n˜ˆ now reads

(81) 2 1 2 n˜ˆ =c ˆ† cˆ ====n ˆ λaˆ† λ∗ aˆ + λ 1ˆ Hˆ = ~ω n˜ˆ + 1ˆ λ 1ˆ , (82) − − | | ⇒ 2 − | | allowing us to express the energy eigenstates of the systen in terms of quasi-particle states:

n (28) (ˆc†) 1 2 n˜ λ ==== 0˜ λ , Hˆ n˜ λ = ~ω n˜ + λ n˜ λ (˜n =0, 1, 2, ) . (83) | i √n! | i | i 2 −| | | i ··· The ground state of the forced linear harmonic oscillator is given by the state without any quasi particles ( quasi-particle vacuum) 0˜ λ , which satisﬁes | iλ ≡| i

(81) vacuum aˆ λ ==== cˆ + λ1ˆ λ ======λ λ . (84) | i | i | i This quasi-particle vacuum is an eigenstate of the original annihilation operatora ˆ belonging to the eigenvalue λ. Such a state is called a coherent state, where the term coherent refers to the fact that all original many-particle states contribute to this state with related phases, as opposed to incoherent situations where the phases are completely random (see Ch. 2). Proof: express the state λ in terms of the original many-particle basis according to | i

∞ (84) ∞ (78) ∞ λ C n === 0 = C aˆ λ1ˆ n ==== C √n n 1 λ n | i ≡ n | i ⇒ n − | i n | − i − | i n=0 n=0 n=0 X X X n+1 basis λCn λ === Cn+1 = = = C0 , ⇒ √n +1 ··· (n + 1)! p

27 where the summation parameter n is increased by one in the term containing n 1 . | − i Next we have to normalize the state λ : | i ′ ∞ n n ∞ 2n 2 λ (λ∗) orth. basis 2 λ 2 2 1 λ λ = C n′ n ======C | | = C exp λ . ≡ h | i | 0| √ √ h | i | 0| n! | 0| | | n,n′ =0 n! n′! n=0 X X By choosing C = exp( λ 2/2) we ﬁnally obtain 0 −| |

∞ λn λ = exp( λ 2/2) n , (85) | i −| | √ | i n=0 n! X which can be rewritten by employing equation (28):

∞ n n 2 λ (ˆa†) 2 λ = exp( λ /2) 0 = exp( λ /2) exp(λaˆ†) 0 . (86) | i −| | √ √ | i −| | | i n=0 n! n! X Indeed we observe that all original many-particle states feature in such a coherent state, with the expectation value for the occupation number and associated quantum mechanical uncertainty given by

(84) 2 2 n¯ λ nˆ λ = λ aˆ†aˆ λ ==== λ λ λ = λ , ≡ h | | i h | | i | | h | i | |

(84) ∆n λ nˆ2 λ λ nˆ λ 2 = λ (â aˆ aâˆ +â aˆ) λ n¯2 ==== √n¯ . (87) ≡ h | | i−h | | i h | † † † | i −

p 2 (75) p Here we used thatn ˆ =a ˆ†aâˆ†aˆ ====a ˆ†aˆ†aâˆ +â†aˆ. The larger the value of λ = √n¯ | | becomes, the larger the average number will be of original oscillator quanta that are present in the ground state of the forced oscillator and the smaller the relative error ∆n/n¯ =1/√n¯ =1/ λ on this number will be! | | The relation between particle number and phase: writing the complex eigenvalue λ as λ = λ exp(iφ ), the corresponding coherent state has the characteristic feature that | | λ the many-particle basis states n occur with phases that are related by exp(inφ ): | i λ n ∞ (86) 2 λ â† 2 iφ λ ==== exp( λ /2) | | exp(inφ ) 0 = exp( λ /2) exp λ e λ aˆ† 0 . | i −| | n! λ | i −| | | | | i n=0 X ∂ iφ (84) i λ =a ˆ† λ e λ λ =a ˆ† λ λ ====a ˆ†aˆ λ =n ˆ λ . ⇒ − ∂φλ | i | | | i | i | i | i Since the coherent states form a complete set (see remark 1), the following relation between particle number and phase emerges ∂ i =n ˆ − ∂φλ

28 in the coherent-state representation. This resembles the quantum mechanical relation between conjugate pairs of coordinates and momenta, such as the coordinate x and the corresponding momentum p for which we have the position-space relationp ˆ = i~∂/∂x. x x − As such we expect to ﬁnd a kind of particle number–phase uncertainty relation of the form ∆n ∆φ 1/2 , (88) λ ≥ although it is not possible to deﬁne a unique phase operator in QM in view of the fact that

φλ is defined on the interval [0, 2π) only. For a fixed-n state we would therefore expect to have a completely undetermined phase. Actually we have already made use of this in 1.2 by exploiting the freedom to choose the overall phase of the occupation-number basis § states. For an arbitrary coherent state we expect to find a compromise between uncertainty in particle number and uncertainty in phase (see remark 2)

Pseudo-classical properties of the coherent state |λi: we know that

k l k l k l λ (â†) (â) λ = (λ∗) (λ) = λ aˆ† λ λ aˆ λ h | | i h | | i h | | i for normal-ordered operators, seemingly bypassing quantum mechanical commutation relations by allowing the operators to be effectively replaced by complex parameters. Well, hold your horses this classical behaviour does not persist once we consider operators ··· that are not normal ordered, such as

2 λ aâˆ† λ = λ (â†aˆ + 1)ˆ λ = λ +1 = λ aˆ λ λ aˆ† λ +1 . h | | i h | | i | | h | | ih | | i The commutation relations lead to relative corrections of 1/ λ 2 = (1/n¯) to classi- O | | O cal behaviour. In the high occupation-number limit λ 1 this correction is suppressed | | ≫ and a classical situation is approached that allows you to effectively replace the relevant many-particle operators by complex numbers in the coherent-state expectation value. Moreover, it will be shown in exercise 6 that the relative overlap between different coherent states decreases substantially in that case (see also remark 2), allowing the coherent states to be viewed as pseudo classical states consisting of an almost fixed, very large number of oscillator quanta.

Unitary transformation: ﬁnally it will be shown in exercise 6 that a coherent state can be obtained from the ground state of the linear harmonic oscillator. As expected this will be achieved by means of a unitary transformation in Fock space:

λ = exp(λaˆ† λ∗aˆ) 0 . (89) | i − | i Coherent states are particularly useful in describing electromagnetic radiation that is produced coherently through stimulated emission. For instance, masers and lasers fall into this category as well as electromagnetic ﬁelds produced by

29 classical electric currents (see Ch.4). Coherent states also feature prominently in the description of systems with macroscopically occupied 1-particle states, such as Bose–Einstein condensates (see Ch.2) and classical electromagnetic ﬁelds (see Ch.4).

Remark 1: overcomplete set (no exam material). The collection of all coherent states

∞ λn λ = exp( λ 2/2) n : λ C (90) | i −| | √ | i ∈ n=0 n! X forms an overcomplete set. To this end we write the complex eigenvalue λ in terms of polar coordinates in the complex plane as λ = λ exp(iφ ). That allows us to derive the | | λ following generalized completeness relation for coherent states:

′ ∞ ∞ ∞ ∞ ∞ n n 2 λ (λ∗) d(Reλ) d(Imλ) λ λ = d(Reλ) d(Imλ) exp( λ ) n n′ | ih | −| | ′ √n!n ! | ih | Z Z Z Z n,n =0 ′ −∞ −∞ −∞ −∞ X 2π δnn′ 2π ∞ ∞ ′ polar coord. n n′ n+n +1 2 ======| ih | d λ λ exp( λ ) z dφλ exp}|iφλ[n n′]{ ′ √n!n ! | || | −| | − n,n =0 ′ Z Z X 0 0

2 ∞ z = λ ∞ n n ∞ compl. ======| | π | ih | dz zn exp( z) = π n n ==== π1ˆ , (91) n! − | ih | n=0 n=0 X Z0 X where it is used that ∞ dz zn exp( z) = n! for n = 0, 1, . An arbitrary many- 0 − ··· particle state is thereforeR readily expressible in terms of coherent states!

Remark 2: two more pseudo-classical properties (no exam material). A coherent state is also an example of a quantum state with minimal uncertainty that is divided equally over position and momentum operators. By combining the results on the previous page with equation (74), it can be proven straightforwardly that

2 2 2 1 2 1 2 1 (∆P ) = λ Pˆ λ λ Pˆ λ = λ (â aˆ†) λ + λ (â aˆ†) λ = , h | | i−h | | i − 2 h | − | i 2 h | − | i 2 2 2 2 1 2 1 2 1 (∆Q) = λ Qˆ λ λ Qˆ λ = + λ (â +â†) λ λ (â +â†) λ = , h | | i−h | | i 2 h | | i − 2 h | | i 2

(72) which indeed yields (∆x)(∆px) ==== ~(∆P )(∆Q) = ~/2. The quantum mechanical expectation values and uncertainties in the (Q,P )-plane are illustrated schematically in the picture displayed below. From the quantum mechanical uncertainties (i.e. the grey

30 circles with radius 1/√2 ) it can be read oﬀ that for increasing λ both the relative | | uncertainty in λ and the absolute uncertainty in the phase φ get smaller. This is | | λ in marked contrast to the case λ = 0, for which the phase φλ is completely undetermined! As will be shown in exercise 6, for λ 1 a classical situation is approached with | | ≫ substantially decreased relative overlap between coherent states. In addition, upon close inspection we indeed observe the anticipated uncertainty relation between particle number (87) and phase: √2 λ ∆φ 1/√2 === ∆n ∆φ 1/2. | | λ ≥ ⇒ λ ≥

P P √2 λ | | ∆φλ √2 Imλ

φλ Q Q √2 Reλ

λ = 0 : 0 λ = 0 : λ | i 6 | i

We ﬁnalize by noting that a coherent state is the quantum state that comes closest in approximating the classical motion of a non-interacting linear harmonic oscillator. Equa- tion (65) tells us that the annihilation operatora ˆ of a linear harmonic oscillator system is given in the Heisenberg picture by

aˆ (t)=ˆa exp( iωt) aˆ (t) λ = λ exp( iωt) λ = λ exp i[φ ωt] λ , H − ⇒ H | i − | i | | λ − | i implying that

λ Qˆ (t) λ = √2 Re λ exp( iωt) and λ Pˆ (t) λ = √2 Im λ exp( iωt) h | H | i − h | H | i − indeed reproduce the time-dependent motion of a linear harmonic oscillator with a minimum of quantum mechanical uncertainty.

1.6.3 Superﬂuidity

As a second application of the concept of quasi particles, we consider the quantum- mechanical low-temperature phenomenon of superfluidity. The explicit example that will be investigated will teach us that a low-energy spectrum of excitations in a bosonic many- particle system can change radically upon switching from a non-interacting (ideal-gas) scenario to a scenario in which the particles repel each other weakly. Before we go into the details, first the concept of superfluidity will be introduced.

31 We speak of a superfluid flow inside a medium object at rest if it is not subject to energy leakage through friction. By means of an elegant argument (formulated by L.D. Landau in 1941) it can be established when a superfluid flow is possi- ~u ble. To this end we consider a fluid that flows ~u at uniform velocity ~u past a macroscopic ob- − ject. Next we switch to a frame of reference in which the fluid is at rest (at absolute zero of temperature) and the object moves at veloc- fluid at rest ity ~u . Friction (viscosity) occurs only then, − when the object can convert part of its kinetic momentum and energy into excitations in the fluid. Let’s have a closer look at such an excitation with momentum ~p and energy ǫ(~p ). The conserved total momentum and energy of the combined system, consisting of the fluid and the macroscopic object with mass M, is given by

1 2 1 2 P~ = M ~u = ~p + M ~u′ and E = M ~u = ǫ(~p ) + M ~u′ , − 2 2

with ~u′ being the velocity of the object after exciting the fluid. By merging both equations one obtains ~p 2 ǫ(~p )+ ~p ~u = . · − 2M From this it follows that the excitation is energetically not possible if the flow velocity of the fluid satisfies the condition ǫ(~p ) u ~u < . ≡ | | ~p | | If the spectrum of fluid excitations ǫ(~p ) displays a minimum (critical velocity)

ǫ(~p ) uc = > 0 , (92) ~p min | |

then superﬂuidity can set in at velocities u

We can distinguish a few special shapes (dispersion relations) of the excitation spectrum.

ǫ(~p )= ~p 2/(2m): this corresponds to the energy spectrum of free non-relativistic • particles. Example: a gas consisting of non-interacting massive particles. In that case ǫ(~p )/ ~p = ~p /(2m) does not possess a positive minimum and according to | | | | Landau’s criterion no superﬂuidity can occur.

32 ǫ(~p )= c ~p : this corresponds to the energy spectrum of free massless quanta. • s | | Example: quantized wave phenomena, such as sound waves in a fluid or crystal (see exercise 5). In that case ǫ(~p )/ ~p equals the speed of propagation c of the waves | | s inside the medium. According to Landau’s criterion, superfluidity can occur for flow

velocities u

The excitation spectrum has a finite energy gap • ∆ = lim ǫ(~p ) between the ground state and the ~p 0 ǫ(~p )/ ~p ǫ(~p ) first excited| |→ state. Example: the energy gap that | | features in the energy spectrum of superconductors as a result of attractive interactions between special pairs (Cooper pairs) of conduction electrons. In that case ǫ(~p )/ ~p clearly displays a minimum and super- | | fluidity can occur if the corresponding flow velocity is low enough. 0 ~p | |

1.6.4 Superﬂuidity for weakly repulsive spin-0 bosons (part 1)

As an example we investigate the low-energy excitation spectrum of weakly repulsive bosons. To this end we consider a system consisting of a very large, constant number N of identical spin-0 particles with mass m. The particles are contained inside a large cube with edges L and periodic boundary conditions, giving rise to a discrete momentum spectrum: ~p = ~~k : k = 0, 2π/L, 4π/L, . (93) { x,y,z ± ± ···} In addition we assume the temperature to be low enough for the system to be eﬀectively in the ground state, i.e. eﬀectively we are dealing with a T = 0 system.2

Without interactions among the particles: the total kinetic energy operator of the non-interacting identical-particle system is diagonal in the momentum representation:

~2~k 2 1 ~2~k 2 Tˆ = aˆ† aˆ = aˆ† aˆ +a ˆ† aˆ , (94) tot ~k ~k ~k ~k ~k ~k 2m 2 2m − − ~k ~k =~0 X X6 with kinetic-energy eigenvalues

~2~k 2 E(0) = n(0) . (95) 2m ~k X~k 2The relevant details regarding temperature dependence and regarding the quantization aspects of the container can be found in chapter 2.

33 The second expression for Tˆtot is given for practical purposes only (see below). It exploits the symmetry of the momentum summation under inversion of the momenta. By adding the total momentum operator ˆ 1 P~ = ~~k aˆ† aˆ = ~~k aˆ† aˆ aˆ† aˆ , (96) tot ~k ~k ~k ~k ~k ~k 2 − − − ~k ~k =~0 X X6 we can readily read oﬀ the particle interpretation belonging to the creation and annihilation operators used. Particles with energy ~2~k 2/(2m) and momentum ~~k are created by (0) aˆ† , annihilated bya ˆ~ and counted byn ˆ~ =ˆa† aˆ~ . The occupation number n indicates ~k k k ~k k ~k how many particles can be found in the given momentum eigenstate in the absence of mutual interactions. The ground state of the non-interacting N-particle system automatically has n(0) = N and n(0) = 0. ~0 ~k =~0 6

Including a weak repulsive interaction among the particles: in analogy with exercise 4, the inclusion of weak pair interactions that depend exclusively on the distance between the two particles will lead to an additive many-particle interaction term of the form ˆ 1 V = (q)ˆa~† aˆ~† ′ aˆ~k ′+~q aˆ~k ~q , (97) 2 U k k − ′ ~k,X~k ,~q where 1 (q) ( ~q ) = d~r (r) exp( i~q ~r ) (98) U ≡ U | | V U − · VZ is the Fourier transform of the spatial pair interaction per unit of volume. As argued in exercise 4, the pair momentum of the interacting particles remains conserved under the interaction (owing to translational symmetry) and (q) merely depends on the absolute U value of ~q (owing to rotational symmetry).

What do we expect for the energy eigenstates in the ineracting case?

First of all the state n~0 = N, n~k =~0 = 0 will no longer be the ground state of the • | 6 i interacting many-particle system. By applying Vˆ we observe

ˆ 1 V n~0 = N, n~k =~0 =0 = (q)ˆa~q† aˆ† ~q aˆ~0 aˆ~0 n~0 = N, n~k =~0 =0 | 6 i 2 U − | 6 i X~q 1 = N(N 1) (0) n~0 = N, n~k =~0 =0 2 − U | 6 i 1 + N(N 1) (q) n~0 = N 2, n~q = n ~q =1, other n~ =0 . 2 − U | − − k i ~q =~0 p X6 Therefore we expect the true ground state to receive explicit (small) contributions from pairs of particles that occupy excited 1-particle eigenstates with opposite mo-

mentum. In that way the net momentum P~tot = ~0 indeed remains unaﬀected.

34 In the non-interacting case the low-energy N-particle excitations involve just a few • particles in excited 1-particle states with (h/L) momenta. Since all particles then O come with (L) de Broglie wavelengths, we expect the quantum mechanical inﬂu- O ence of the particles to extend across the entire system. Upon including repulsive eﬀects, we expect a single excitation to involve more than just bringing a single particle into motion and hence to require more energy than in the non-interacting case.

Approximations for weakly repulsive dilute Bose gases (Bogolyubov, 1947): if the pair interaction is suﬃciently weak and repulsive, we expect for the low-energy N-particle states that still almost all particles occupy the 1-particle ground state, i.e. N nˆ N . This allows us to apply two approximation −h ~0i ≪ steps to simplify the many-particle interaction term, which hold as long as the gas is suﬃciently dilute to avoid too many particles from ending up in excited 1-particle states.

Approximation 1: for obtaining the low-energy N-particle states the eﬀect from interactions among particles in excited 1-particle states can be neglected. This boils down to only considering interaction terms with two or more creation and annihilation operators that belong to the 1-particle ground state:

2 nˆ nˆ~ nˆ nˆ nˆ ~0 − 0 ~0 ~k ~0 ˆ 1 1 V (0) aˆ~† aˆ~† aˆ~0 aˆ~0 + (0) aˆ~† aˆ~0 aˆ~† aˆ~k + aˆ~† aˆ~0 (q) aˆ~q† aˆ~q +a ˆ† ~q aˆ ~q ≈ 2 U 0 0 U 0 k 2 0 U − − ~k =~0 ~q =~0 z }| { z}|{ X6 z }| { z}|{ X6 1 + (q) aˆ~0 aˆ~0 aˆ~q† aˆ† ~q +a ˆ~† aˆ~† aˆ~q aˆ ~q . 2 U − 0 0 − ~q =~0 X6

The ﬁrst term corresponds to the conﬁguration ~k = ~k ′ = ~q = ~0 , where all creation • and annihilation operators refer to the 1-particle ground state.

The remaining terms cover situations where only two out of three momenta vanish. • The second term corresponds to the conﬁgurations ~k = ~q = ~0 and ~k ′ = ~q = ~0 ,

the third term to ~k ′ = ~k ~q = ~0 and ~k = ~k ′ + ~q = ~0, and the last term to − ~k ~q = ~k ′ + ~q = ~0 and ~k = ~k ′ = ~0. − ˆ Subsequently the number operatorn ˆ~0 can be replaced everywhere by N nˆ~q and the − ~q =~0 total number of particles N can be taken as ﬁxed and very large: P6

ˆ N(N 1) 1 V − (0) + (q) N aˆ~q† aˆ~q + N aˆ† ~q aˆ ~q +a ˆ~0 aˆ~0 aˆ~q† aˆ† ~q +a ˆ~† aˆ~† aˆ ~q aˆ~q . ≈ 2 U 2 U − − − 0 0 − ~q =~0 X6

35 Approximation 2, traditional approach: in Bogolyubov’s approach it is used that the operatorsa ˆ aˆ anda ˆ† aˆ† can be eﬀectively replaced by N when applied to the ~0 ~0 ~0 ~0 lowest-energy N-particle states. In principle this could involve extra phase factors e2iφ0

2iφ0 and e− , however, these can be absorbed into a redefinition of the remaining creation and annihilation operators. This approach suggests that we are dealing with an approximately classical situation, where the fact that the operatorsa ˆ anda ˆ† do not ~0 ~0 commute only affect the considered N-particle states in a negligible way (as if they were coherent states with λ 1). This assumption is plausible, bearing in mind that | | ≫ √N n √N if N n. As a result, magnitude-wise the action ofa ˆ anda ˆ† on − ≈ ≫ ~0 ~0 states with N nˆ N will be effectively the same. In this way the following effective −h ~0i≪ approximation is obtained for the total Hamilton operator Hˆtot = Tˆtot + Vˆ applicable to the lowest lying energy eigenstates:

~2 2 ˆ N(N 1) 1 q Htot − (0) + + N (q) aˆ~q† aˆ~q +a ˆ† ~q aˆ ~q ≈ 2 U 2 2m U − − ~q =~0 X6 1 + N (q) aˆ~q† aˆ† ~q +a ˆ ~q aˆ~q . (99) 2 U − − ~q =~0 X6 Owing to this second approximation the total number of particles is no longer conserved under the interactions. The justiﬁcation for such an approach is purely thermodynamic by nature (see Ch. 2): “the physical properties of a system with a very large number of particles do not change when adding/removing a particle”. Since approximations are unavoidable for the description of complex many-particle systems, non-additive quantities of this type are ubiquitous in condensed-matter and low-temperature physics. Note, however, that the total momentum of the many-particle system remains conserved under the interaction, since each term in Hˆ adds as much momentum as it subtracts.

Approximation 2, but this time conserving particle number (based on the bachelor thesis by Leon Groenewegen): in order to avoid that particle number is not conserved during the second approximation step we can again exploit approximation 1 and write N aˆ aˆ† . Without loss of accuracy this allows to rewrite the total Hamilton operator ≈ ~0 ~0 Hˆtot = Tˆtot + Vˆ as

~2 2 ˆ N(N 1) 1 q Htot − (0) + + (q) aˆ~0 aˆ~† aˆ~q† aˆ~q +a ˆ~0 aˆ~† aˆ† ~q aˆ ~q ≈ 2 U 2 2mN U 0 0 − − ~q =~0 X6 1 + (q) aˆ~0 aˆ~0 aˆ~q† aˆ† ~q +a ˆ~† aˆ~† aˆ ~q aˆ~q 2 U − 0 0 − ~q =~0 X6

36 ~2 2 N(N 1) 1 q ˆ ˆ ˆ ˆ − (0) + + N (q) b~q† b~q + b† ~q b ~q ≡ 2 U 2 2m U − − ~q =~0 X6 1 ˆ ˆ ˆ ˆ + N (q) b~q† b† ~q + b ~q b~q , (100) 2 U − − ~q =~0 X6 where the operators

aˆ aˆ† aˆ† aˆ aˆ† aˆ aˆ aˆ† ~0 ~q ~q ~0 ~0 ~q ~q ~0 ˆb† = = and ˆb = = (101) ~q =~0 ~q =~0 6 √N √N 6 √N √N ˆ have a much clearer physical interpretation. By means of b~q† a particle is excited from the 1-particle ground state to the 1-particle state with momentum ~~q = ~0, whereas ˆb 6 ~q describes the de-excitation of an excited particle to the 1-particle ground state. The total number of particles is not aﬀected in this way! However, the total Hamilton operator is identical to equation (99), with the non-additive character of Hˆtot simply following from the fact that the interaction can both excite pairs of particles from the 1-particle ground state and make them fall back to the ground state. As a result of the macroscopic ˆ ˆ occupation of the 1-particle ground state, the operators b~q† and b~q approximately behave as ordinary creation and annihilation operators:

† ′ 1 nˆ~0 aˆ~q aˆ~q ˆb , ˆb ′ = 0 and ˆb , ˆb† ′ = aˆ† aˆ , aˆ† ′ aˆ = aˆ , aˆ† ′ aˆ , aˆ† ′ . ~q ~q ~q ~q N ~0 ~q ~q ~0 ~q ~q N − N ≈ ~q ~q More details can be found in the bachelor thesis by Leon Groenewegen.

In analogy with 1.6.2 it is opportune to switch now to a quasi-particle de- § scription that turns the approximated interacting system into a non-interacting quasi-particle system, with a corresponding total Hamilton operator that is both additive and diagonal (i.e. decoupled).

ˆ In this quasi-particle description we expect that we have to combine the operator pairs b~q† ˆ ˆ ˆ and b ~q as well as b~q and b† ~q , since − − both operators within such a pair describe the same change in momentum and there- • fore aﬀect the total momentum of the many-particle system in the same way;

ˆ ˆ ˆ ˆ the quasi-particle number operators will then generate the correct terms b~q† b~q , b ~q b† ~q , • ˆ ˆ ˆ ˆ − − b~q† b† ~q and b ~q b~q . − − How to ﬁnd the correct quasi-particle description will be addressed in the next intermezzo.

37 1.6.5 Intermezzo: the Bogolyubov transformation for bosons

Consider a many-particle system consisting of identical bosons that can occupy two fully speciﬁed 1-particle quantum states q and q , such as the momentum states ~q and | 1i | 2i | i ~q in 1.6.4. The corresponding creation and annihilation operators are given bya ˆ† , aˆ† |− i § 1 2 anda ˆ , aˆ . The corresponding Fock space is spanned by the basis states n , n as given 1 2 | 1 2i in equation (28), where n1,2 represents the number of identical particles in each of the two 1-particle quantum states. Next we consider a non-additive Hamilton operator of the form

Hˆ = E (â† aˆ +â† aˆ ) + ∆(â† aˆ† +â aˆ ) (E > 0 and ∆ IR) . (102) 1 1 2 2 1 2 2 1 ∈ Evidently, the total number of particles is not conserved by such an operator, as the occupation number of each of the two quantum states is raised by one or lowered by one in the ∆ terms. However, by means of a so-called Bogolyubov transformation it can be cast into an additive form up to a constant term.3 Such a transformation has the generic form

cˆ u aˆ + v aˆ† , cˆ u aˆ + v aˆ† (u and v IR) . (103) 1 ≡ 1 1 1 2 2 ≡ 2 2 2 1 1,2 1,2 ∈

The real constants u1 ,u2 , v1 and v2 will be chosen in such a way that the operatorsc ˆ1†,2 and cˆ1,2 satisfy the same bosonic commutator algebra asa ˆ1†,2 anda ˆ1,2 . This will allow us to

formulate a new particle interpretation, wherec ˆ1†,2 andc ˆ1,2 describe the creation and an-

nihilation of quasi particles. In order to guarantee that botha ˆ1†,2 , aˆ1,2 andc ˆ1†,2 , cˆ1,2 satisfy bosonic commutation relations, the following conditions should hold:

u v v u = 0 and u2 v2 = u2 v2 = 1 1 2 − 1 2 1 − 1 2 − 2 u = + u , v = + v and u2 v2 = 1 (104) ⇒ 1 2 1 2 1 − 1 or u = u , v = v and u2 v2 = 1 . 1 − 2 1 − 2 1 − 1

Proof: the commutation relations cˆ1, cˆ1 = cˆ2, cˆ2 = cˆ1, cˆ2† = 0 follow directly from the bosonic commutation relations fora ˆ1†,2 anda ˆ1,2 . The indicated conditions for u1,2 and v1,2 then simply follow from the fact that the other commutators

(103) (26) cˆ , cˆ ==== u aˆ + v aˆ† ,u aˆ + v aˆ† ==== (u v v u )1ˆ , 1 2 1 1 1 2 2 2 2 1 1 2 − 1 2

(103) (26) 2 2 cˆ , cˆ† ==== u aˆ + v aˆ† ,u aˆ† + v aˆ ==== (u v )1ˆ , 1 1 1 1 1 2 1 1 1 2 1 − 1

(103) (26) 2 2 cˆ , cˆ† ==== u aˆ + v aˆ† ,u aˆ† + v aˆ ==== (u v )1ˆ , 2 2 2 2 2 1 2 2 2 1 2 − 2 have to satisfy the usual bosonic commutation relations. 3 If the states q1 and q2 carry opposite quantum numbers, such as the momentum in the states ~q | i | i | i and ~q in 1.6.4, then both Hˆ and the Bogolyubov transformation conserve these quantum numbers. |− i § 38 The sign in this transformation can be chosen freely. Usually one chooses the plus sign, resulting in the following generic form for a bosonic Bogolyubov transformation:

cˆ u aˆ + v aˆ† , cˆ u aˆ + v aˆ† (105) 1 ≡ 1 1 1 2 2 ≡ 1 2 1 1 with inverse

aˆ = u cˆ v cˆ† , aˆ = u cˆ v cˆ† . (106) 1 1 1 − 1 2 2 1 2 − 1 1

In the literature one usually opts for a parametrization of u1 and v1 in terms of a real parameter η according to u1 = cosh η and v1 = sinh η, which automatically satisﬁes the condition u2 v2 = 1. 1 − 1 Bringing the non-additive Hamilton operator of equation (102) in additive form. To this end we consider two combinations of quasi-particle number operators. First of all (105) cˆ† cˆ cˆ† cˆ ==== (u aˆ† + v aˆ )(u aˆ + v aˆ† ) (u aˆ† + v aˆ )(u aˆ + v aˆ† ) 1 1 − 2 2 1 1 1 2 1 1 1 2 − 1 2 1 1 1 2 1 1

(26) 2 2 ==== u (ˆa† aˆ aˆ† aˆ ) v (ˆa aˆ† aˆ aˆ†) 1 1 1 − 2 2 − 1 1 1 − 2 2

(26) 2 2 (104) ==== (u v )(â† aˆ aˆ† aˆ ) ====a ˆ† aˆ aˆ† aˆ . (107) 1 − 1 1 1 − 2 2 1 1 − 2 2 This expression tells us that certain quantum numbers are conserved under the transition from particles to quasi particles, provided that these quantum numbers take on opposite values in the states q and q (see the footnote on p.38). In 1.6.6 we will use this | 1i | 2i § property to conserve the momentum quantum numbers and thereby the total momentum while performing the transformation, as anticipated in 1.6.4. Secondly we have § ˆ 2 2 ˆ cˆ1† cˆ1 +ˆc2† cˆ2 + 1 = (u1 + v1)(â1† aˆ1 +â2† aˆ2 + 1) + 2u1v1 (â1† aˆ2† +â2 aˆ1) . (108) This implies that for ∆ < E the non-additive Hamilton operator Hˆ in equation (102) | | can be rewritten as

Hˆ = E (â† aˆ +â† aˆ + 1)ˆ + ∆(â† aˆ† +â aˆ ) E 1ˆ 1 1 2 2 1 2 2 1 −

= √E2 ∆2 (ˆc† cˆ +ˆc† cˆ + 1)ˆ E 1ˆ . (109) − 1 1 2 2 − 2 2 Proof: based on equation (108) we are looking for a factor C such that C(u1 + v1) = E and 2Cu1v1 = ∆. This implies that E and C should have the same sign and that

(104) E,C> 0 E2 ∆2 = C2(u2 v2)2 ==== C2 ======∆ < E and C = √E2 ∆2 . − 1 − 1 ⇒ | | − As promised at the start of this intermezzo, the Hamilton operator has been cast into a form that consists exclusively of a unit operator and number operators for quasi particles. In exercise 7 it will be shown that the corresponding ground state (i.e. the state without quasi-particle excitations) is comprised of coherently created pairs of particles. In 1.7.2 § the fermionic version of all this will be derived by employing a similar procedure.

39 1.6.6 Superﬂuidity for weakly repulsive spin-0 bosons (part 2)

The approximation (100) for the total Hamilton operator at the end of 1.6.4 is exactly of § the form discussed in the previous intermezzo. Therefore, an appropriate set of Bogolyubov transformations can be performed to recast the Hamilton operator in the additive form

~2 2 ˆ N(N 1) 1 q 1 H − (0) + N (q) ǫ~q + ǫ~q (ˆc~q† cˆ~q +ˆc† ~q cˆ ~q) ≈ 2 U − 2 2m U − 2 − − ~q =~0 ~q =~0 X6 X6 N(N 1) 1 ~2q2 = − (0) + N (q) ǫ + ǫ cˆ† cˆ , (110) 2 U − 2 2m U − ~q ~q ~q ~q ~q =~0 ~q =~0 X6 X6 with corresponding quasi-particle excitation spectrum

~2q2 4mN (q) ǫ~q = ǫ ~q = 1 + U . (111) − 2m s ~2q2

For each pair of momenta ~q and ~q with ~q = ~0 a bosonic Bogolyubov transformation − 6 of the type ˆ ˆ ˆ ˆ cˆ~q u~q b~q + v~q b† ~q , cˆ ~q u~q b ~q + v~q b~q† ≡ − − ≡ − should be used, based on the following energy parameters in equations (102) and (109):

1 ~2q2 1 E + N (q) , ∆ N (q) . → 2 2m U → 2 U For the total momentum operator we ﬁnd with the help of approximation 2 on p.37 that

aˆ aˆ† ~ˆ (96) 1 ~ 1 ~ ~0 ~0 Ptot ==== ~q aˆ~q† aˆ~q aˆ† ~q aˆ ~q ~q aˆ~q† aˆ~q aˆ† ~q aˆ ~q (112) 2 − − − ≈ 2 N − − − ~q =~0 ~q =~0 X6 X6 (101) 1 ~ ˆ ˆ ˆ ˆ (107) 1 ~ ~ ==== ~q b~q† b~q b† ~q b ~q ==== ~q cˆ~q† cˆ~q cˆ† ~q cˆ ~q = ~q cˆ~q† cˆ~q . 2 − − − 2 − − − ~q =~0 ~q =~0 ~q =~0 X6 X6 X6

Now we can read oﬀ the new particle interpretation. Quasi particles with energy ǫ~q and ~ momentum ~q are created byc ˆ~q† , annihilated byc ˆ~q and counted byc ˆ~q† cˆ~q . The ground state of the new many-particle system still contains no quanta with momentum ~~q = ~0 . 6 However, both the composition of the ground state in terms of the original particles and the shape (dispersion relation) of the elementary excitation spectrum have changed as a result of the interaction.

40 1) Approximated excitation spectrum for weakly repulsive spin-0 bosons:

For large excitation energies ~2q2 mN (q) the quasi-particle excitation spec- • ≫ |U | trum ǫ ~2q2/(2m)+N (q) hardly diﬀers from the non-interacting spectrum. As ~q ≈ U such, the quasi particles have the same properties as the original bosons.

For small excitation energies ~2q2 mN (q) the quasi-particle excitation spec- • ≪ U trum ǫ ~q N (q)/m ~q N (0)/m changes substantially. The quasi ~q ≈ U ≈ U particles describep massless quanta, i.e.p quantized sound waves inside the considered medium with speed of propagation

ǫ~q N (0) 1 cs = lim = U for (0) = d~r (r) . (113) q 0 ~ → q m U V U r VZ

The low-energy quasi-particle interpretation of the ǫ~q interacting system therefore differs fundamentally from the original particle interpretation of the non- interacting system. Moreover, we have gone from a free-particle system that does not allow for the possi- quadratic bility of superfluidity to an interacting system that can display superfluidity if u

~q | | 2) Approximated ground state for weakly repulsive spin-0 bosons: another thing that has changed substantially is the composition of the ground state of the interacting many- particle system in terms of the original particles (see exercise 7).

In the non-interacting case all particles occupy the 1-particle ground state with mo- • mentum ~0 and energy 0 (i.e. n(0) = N and n(0) = 0). This condensate has spatial ~0 ~k =~0 correlations on all distance scales in view of the corresponding6 macroscopic de Broglie wavelength.

The latter still holds in the interacting case. However, the particle excitations for • ~k = ~0 are replaced by quasi-particle excitations with the same momentum. In the 6 new ground state none of these quasi-particle excitations are occupied (i.e. n N ~k =~0 ≈ and for the quasi particlesn ˜~k =~0 = 0). This situation without excited quasi particles 6 diﬀers markedly from the situation without excited particles. Written in terms of the original particle interpretation the new condensate actually contains particles that are not in the ground state. To be more precise, it contains coherently excited particle pairs with opposite momenta. The increase in kinetic energy is in that case compensated by the decrease in repulsive interaction energy.

41 Remark: if the average spatial pair interaction (0) would have been attractive, i.e. if U (0) < 0, then the ground state would have been unstable. This can be read off directly U from the spectrum (111), which would become complex at very low energies. In that case it would be energetically favourable for the system to have a large number of low-energy particle pairs outside of the 1-particle ground state, which invalidates the assumptions of the previous approach. The fermionic version of such a pairing effect and the ensuing “pair-bonding” instability play a crucial role in the quantum mechanical description of superconductivity. A first exposure to this phenomenon can be found in exercise 9.

1.6.7 The wonderful world of superﬂuid 4He: the two-ﬂuid model

The many-particle system worked out above is often used to model the low-energy excitations 1 rotons ǫ~q kB− phonons in liquid 4He, which becomes superfluid be- ∗ [K] low Tλ =2.18K (P.L. Kapitsa, J.F. Allen and A.D. Misener, 1937). At this point we have to add a critical note. The pair interactions cannot be truly weak, as required by the approximation method, as we are dealing with a fluid rather than a gas. As a consequence, a second branch of ex- 1 ~q [A˚− ] citations enters the spectrum at higher energies, | | lowering the critical velocity for superfluidity. At the absolute zero of temperature 4He is com- fraction helium II pletely superfluid, i.e. no low-energy excitations are excited thermally. At a temperature around 0.9 K a noticable influence of thermally induced excitations sets in. On the temperature inter- fraction helium I val 0

T > Tλ the inﬂuence of the helium II component is negligible. By means of this so-called two-ﬂuid model a few surprising phenomena can be understood.

How to recognize Tλ (P.L. Kapitsa, 1937): a remarkable superﬂuid phenomenon is that the boiling of liquid helium abruptly subsides the moment that the temperature drops be-

low Tλ . Helium I will flow away from any spot where the fluid is locally warmer, thereby carrying away thermal energy, whereas non-thermal helium II will flow towards that spot in

42 order to compensate for the drop in mass density. This superfluid helium II is characterized by an infinitely efficient heat conductance, making it effectively impossible to have a tem-

perature gradient in the ﬂuid. Already for temperatures just below Tλ the heat transport becomes a millionfold more eﬃcient and gas bubbles will have no time to form.

4 Friction for T < Tλ : objects moving through He at temperatures below Tλ experience friction exclusively from the helium I component and not from the helium II component. This allows us to experimentally determine the helium I and helium II fluid fractions (see the picture on the previous page). For temperatures below about 0.9 K the fluid behaves almost entirely as a superfluid, giving rise to a persistent current once the fluid is set in motion (e.g. at higher temperature).

Frictionless flow through porous media: as a result of friction the helium I component can not flow through very narrow capillary channels. However, the superfluid helium II component can do this without the need for a pressure difference between the two ends of such a 4 channel. So, He can flow through porous surfaces if T < Tλ . This flow will be completely frictionless. Such a situation where a selective superfluid flow occurs is called a super leak.

The fontain eﬀect (J.F. Allen and H. Jones, 1938): Fisher and Pickett, Nature 444, 2006 consider an experimental set-up consisting of two containers with 4He that are connected by means of a porous plug. Both containers are cooled down

to the same temperature below Tλ . If the temperature in one of the containers is raised slightly, e.g. by shining a pocket torch on it, then the number of thermally induced excitations will increase in that container. As a result, the helium I component increases at the expense of the helium II component. In order to compensate for the difference in helium II concentration, helium II from the other container will pass through the porous plug. However, a compensatory reverse flow of helium I to the other container, where the helium I concentration is lower, is hampered by friction. Conse- quently, an increased fluid concentration will ac- cumulate in the heated container (heat pump). By providing the heated container with a capillary safety-relief outlet, a spectacular helium fountain (fontain effect) is produced.

43 Creeping helium film: 4He has as an additional property that the mutual van der Waals bonding is weaker than the van der Waals bonding to other atoms. Because of this a 30 nm thick 2-dimensional helium film (Rollin film) attaches itself to the entire wall of a closed helium container. If part of this helium film could flow or drip to a lower level within the container (see picture), then a superfluid helium II flow will start that terminates only when the helium level reaches its energetic optimum everywhere in the container. During this process the helium liquid is seemingly defying gravity.

In case you want to see some video evidence for the bold statements that were made about the weird and wacky world of superﬂuid 4He, then you are advised to have a look at http://www.youtube.com/watch?v=2Z6UJbwxBZI or to do your own YouTube search.

1.7 Examples and applications: fermionic systems

1.7.1 Fermi sea and hole theory

Consider a fermionic many-particle system consisting of a very large, constant number N of electrons with mass m. The electrons are contained inside a large cube with edges L and periodic boundary conditions, giving rise to a discrete momentum spectrum:

~p = ~~k : k = 0, 2π/L, 4π/L, . (114) { x,y,z ± ± ···} Without mutual interactions among the electrons the total kinetic-energy operator of the non-interacting identical-particle system can be written in diagonal form in the momentum representation:

~2~ 2 ˆ k Ttot = aˆ~† aˆ~k,m , (115) 2m k,ms s ~ ms = 1/2 Xk X± with kinetic energy eigenvalues

~2~k 2 E = n~ . (116) 2m k,ms ~ k,mXs By adding the total momentum operator and total spin operator along the z-direction

~ˆ ~~ ~ˆ ~ Ptot = k aˆ~† aˆ~k,m and Stot ~ez = ms aˆ~† aˆ~k,m (117) k,ms s · k,ms s ~ ~ k,mXs k,mXs 44 we can trivially read oﬀ the particle interpretation belonging to the creation and annihilation operators used. Particles with kinetic energy ~2~k 2/(2m), momentum ~~k and

spin component ms~ along the z-axis are created bya ˆ† , annihilated bya ˆ~ and ~k,ms k,ms counted byn ˆ~ =a ˆ† aˆ~ . The occupation number n~ indicates how many parti- k,ms ~k,ms k,ms k,ms cles can be found in the given momentum and spin eigenstate in the absence of mutual interactions.

Ground state : for the ground state of the non-interacting N-electron system this auto- ~ ~ matically implies that n~ = 1 if k kF and n~ = 0 if k >kF . Such a system is k,ms | | ≤ k,ms | | called a completely degenerate electron gas. The fully speciﬁed 1-particle energy levels are ~2 2 occupied from the bottom up by one electron up to the Fermi-energie EF = kF /(2m). In that case

~2~k 2 N = n~ = 2 and Eground = 2 . (118) k,ms 2m ~k,ms ~k k ~k k X | X|≤ F | X|≤ F The corresponding ground state is given by

(0) Ψground = aˆ† aˆ† Ψ aˆ~ Ψground = 0 if ~k > kF . (119) ~k, 1 ~k, 1 k,ms | i 2 − 2 | i ⇒ | i | | ~k k | Y|≤ F This ground state, better known as the Fermi sea , has vanishing total momentum and spin, but a substantial kinetic energy. Even in the ground state the electron gas exerts pressure (see Ch. 2)!

Excited states : the simplest excited states empty ~k2 can be obtained by exciting a single electron ~ in the Fermi sea to an energy that places it k1 kF above the Fermi sea (see picture). As the states inside the Fermi sea are all occupied, occupied excitations are (energetically) easier from energy levels near the edge of the Fermi sea than ground state: electron–hole from lower-lying energy levels. This aspect full Fermi sea excitation plays a decisive role in the thermal properties of fermionic many-particle systems at moderate temperatures (see Ch. 2). An excitation of this type can be represented in creation and annihilation language in the following way:

Ψex =a ˆ† aˆ~ Ψground if ~k1 kF < ~k2 . (120) ~k ,m k1,ms | i 2 s2 1 | i | | ≤ | | Since the excitation creates a hole in the Fermi sea, it also goes under the name of electron–hole excitation. In the quantum mechanical approach of hole theory this aspect

45 is highlighted by employing the Bogolyubov transformations

~ ~ aˆ~ cˆ† if k kF anda ˆ† cˆ† if k > kF (121) k,ms ~k, ms ~k,ms ~k,ms ≡ − − | | ≤ ≡ | | to switch to a description where a hole in the Fermi sea is upgraded to the status of quasi particle with opposite quantum numbers. In that case the Fermi sea constitutes a state without any holes in the sea or excitations above it, i.e. the Fermi sea is the corresponding quasi-particle vacuum state! An excited state of the above type simply boils down to quasi-particle pair creation. The hole-theory approach is particularly useful for the description of electrical conduction and for Dirac’s attempt to set up relativistic QM.

This approach is only possible for fermions: we have actually made use of

Pauli’s exclusion principle, which states that n~ [0, 1], in order to in- k,ms ∈ terchange the role of empty and occupied in the particle interpretation of hole theory. The advantage of doing this is that the Fermi sea (ground state) is the vacuum state within this alternative particle interpretation and that an excited state corresponds to the creation of a quasi-particle pair. In the hole-theory approach conservation of the number of (quasi) particles is sacriﬁced deliberately for achieving ease of use!

Weak interactions among the electrons: for weakly repulsive interactions we can in general use perturbation theory, utilizing the above-mentioned non-interacting situation as unperturbed starting point. According to hole theory the Fermi sea can be treated as vacuum state. For weakly attractive interactions the situation changes substantially. In exercise 9 it will be shown how the presence of the Fermi sea of occupied states can give rise to the so-called Cooper instability. This entails that two electrons that occupy states above a Fermi sea of occupied states will always form a “bound” pair with ﬁnite binding energy as long as their mutual interaction is attractive. In contrast to a 1-particle bound state in a potential well, this many-particle binding eﬀect occurs irrespective of the strength of the interaction. Moreover, it cannot be described by means of perturbation theory.

1.7.2 The Bogolyubov transformation for fermions

Also a fermionic version of the Bogolyubov transformation of 1.6.5 exists. To this end § we consider exactly the same 2-level system as described on page 38. Only after equation (103) things will start to deviate. So, also here the transformation will be deﬁned by

cˆ1 = u1 aˆ1 + v1 aˆ2† andc ˆ2 = u2 aˆ2 + v2 aˆ1† .

In contrast to the bosonic case, botha ˆ1†,2 , aˆ1,2 andc ˆ1†,2 , cˆ1,2 have to satisfy fermionic anticommutation relations. That implies the following conditions for the real constants

46 u1,2 and v1,2 :

2 2 2 2 u1v2 + v1u2 = 0 and u1 + v1 = u2 + v2 = 1

u = + u , v = v and u2 + v2 = 1 (122) ⇒ 1 2 1 − 2 1 1 or u = u , v = + v and u2 + v2 = 1 . 1 − 2 1 2 1 1

Proof: the anticommutation relations cˆ1, cˆ1 = cˆ2, cˆ2 = cˆ1, cˆ2† = 0 follow directly from the fermionic anticommutation relations fora ˆ1†,2 and a ˆ1,2 . The indicated conditions for u1,2 and v1,2 then simply follow from the fact that the other anticommutators

(103) (27) ˆ cˆ1, cˆ2 ==== u1 aˆ1 + v1 aˆ2† ,u2 aˆ2 + v2 aˆ1† ==== (u1v2 + v1u2)1 , (103) (27) 2 2 ˆ cˆ1, cˆ1† ==== u1 aˆ1 + v1 aˆ2† ,u1 aˆ1† + v1 aˆ2 ==== (u1 + v1)1 , (103) (27) 2 2 ˆ cˆ2, cˆ2† ==== u2 aˆ2 + v2 aˆ1† ,u2 aˆ2† + v2 aˆ1 ==== (u2 + v2)1 , have to satisfy the usual fermionic anticommutation relations. The sign in this transformation can be chosen freely. Usually one chooses the upper sign (i.e. u = u and v = v ), 1 2 1 − 2 resulting in the following generic form for a fermionic Bogolyubov transformation:

cˆ u aˆ + v aˆ† , cˆ u aˆ v aˆ† (123) 1 ≡ 1 1 1 2 2 ≡ 1 2 − 1 1 with inverse

aˆ = u cˆ v cˆ† , aˆ = u cˆ + v cˆ† . (124) 1 1 1 − 1 2 2 1 2 1 1

In the literature one usually opts for a parametrization of u1 and v1 in terms of a real parameter θ according to u1 = cos θ and v1 = sin θ, which automatically satisﬁes the 2 2 condition u1 + v1 = 1.

Special case: if u1 = u2 = 0, then the signs of both v1 and v2 can be chosen independently. Usually one will choose v = v = 1, as we did in 1.7.1 while describing the quasi 1 2 § particles in hole theory.

The fermionic quasi-particle vacuum : removing a particle from a particular state is in the fermionic case equivalent with the creation of a hole in the occupation of that state. The type of quasi particle we are dealing with here is therefore a linear combination of a particle and a hole. This type of quasi particle is used abundantly in condensed-matter physics for describing systems of interacting fermions such as conduction electrons. In exercise 8 the fermionic quasi-particle vacuum 0˜ will be expressed in terms of the original basis states | i

47 n , n , resulting in the compact expression | 1 2i up to a phase factor 0˜ ======u 0, 0 v 1, 1 = u v aˆ† aˆ† 0, 0 . (125) | i 1 | i − 1 | i 1 − 1 1 2 | i Bringing the non-additive Hamilton operator of equation (102) in additive form. To this end we consider two combinations of quasi-particle number operators. First of all

(123) cˆ† cˆ cˆ† cˆ ==== (u aˆ† + v aˆ )(u aˆ + v aˆ†) (u aˆ† v aˆ )(u aˆ v aˆ†) 1 1 − 2 2 1 1 1 2 1 1 1 2 − 1 2 − 1 1 1 2 − 1 1

(27) 2 2 ==== u (ˆa† aˆ aˆ† aˆ ) v (ˆa aˆ† aˆ aˆ† ) 1 1 1 − 2 2 − 1 1 1 − 2 2

(27) 2 2 (122) ==== (u + v )(ˆa† aˆ aˆ† aˆ ) ====a ˆ† aˆ aˆ† aˆ . (126) 1 1 1 1 − 2 2 1 1 − 2 2 This expression tells us again that certain quantum numbers are conserved under the transition from particles to quasi particles, provided that these quantum numbers take on opposite values in the states q and q . | 1i | 2i Secondly we have

2 2 cˆ† cˆ +ˆc† cˆ 1ˆ = (u v )(â† aˆ +â† aˆ 1)ˆ + 2u v (â† aˆ† +â aˆ ) . (127) 1 1 2 2 − 1 − 1 1 1 2 2 − 1 1 1 2 2 1 The non-additive Hamilton operator Hˆ in equation (102) can then be rewritten as

Hˆ = E (â† aˆ +â† aˆ 1)ˆ + ∆(â† aˆ† +â aˆ ) + E 1ˆ 1 1 2 2 − 1 2 2 1

= √E2 + ∆2 (ˆc† cˆ +ˆc† cˆ 1)ˆ + E 1ˆ . (128) ± 1 1 2 2 − Proof: based on equation (127) we are looking for a factor C such that C(u2 v2) = E 1 − 1 2 2 2 2 2 2 (122) 2 and 2Cu1v1 = ∆. From this it follows that E + ∆ = C (u1 + v1 ) ==== C , where the sign of C is determined by the sign of (u2 v2)/E . 1 − 1 As desired, the Hamilton operator has been cast into a form that consists exclusively of a unit operator and number operators for quasi particles. For the description of electron holes in 1.7.1 we have actually used a special version of this wherec ˆ =a ˆ† enc ˆ =a ˆ† . § 1 2 2 1 In that case E (â† aˆ +â† aˆ ) = E (â aˆ† +â aˆ† 2) = E (ˆc† cˆ +ˆc† cˆ 2). From the 1 1 2 2 − 1 1 2 2 − − 2 2 1 1 − perspective of the Fermi energy, the creation of a hole in the Fermi sea results in a negative energy contribution. As noted in the proof given above, the sign of the energy of the quasi particle is fixed by the sign of (u2 v2)/E, which indeed indicates whether the considered 1 − 1 type of quasi particle is more particle than hole (positive sign) or more hole than particle (negative sign).

48 2 Quantum statistics

In this chapter the concept of mixed quantum mechanical ensembles will be introduced. Based on this concept the various types of quantum statistics will be derived, which govern many-particle systems that are in thermal equilibrium. Similar material can be found in Griﬃths (the quantum statistics part in Ch.5), Merzbacher (Ch.15,16,22) and Bransden & Joachain (Ch.10,14).

Up to now we have been working with pure quantum mechanical ensembles belonging to a pure quantum state, i.e. a collection of identical, independent, identically prepared systems that can be described by a single state function ψ . This state function can be determined | i up to a phase by performing a complete measurement on a complete set of commuting observables. The systems that make up the ensemble are independent of each other and can consist of individual particles, interacting particle clusters (such as atoms/molecules) or even entire many-particle systems (such as gases). These pure quantum states can be realized experimentally by employing dedicated ﬁlters, such as a Stern–Gerlach ﬁlter for selecting a pure spin state.

Question: What happens if not all the degrees of freedom that are relevant for the considered systems are taken into account, thereby leaving out certain quantum information?

This situation actually reﬂects the unavoidable

reality of the quantum mechanical world that we cooled gas reservoir: experience. During a quantum experiment the stud- (1023) particles ied system is for instance prepared by filters and O subsequently subjected to the environment of the experimental set-up (gases, electromagnetic fields, etc.). All these environmental influences caused our favourite system by the contact with the macroscopic outside world in fact determine the properties of the quantum

mechanical ensemble that eventually will be mea- cooling elements sured. Examples of this are quantum systems that have their themodynamical properties ﬁxed by the contact with a macroscopic gas reservoir present within the measurement apparatus, or a particle beam that is produced by a particle source and subsequently passes through a set of ﬁlters. It is not feasible to determine the state function of the entire experimental apparatus by means of a complete measurement. The macroscopic outside world simply has way too many degrees of freedom. In such situations the only practical approach is to explicitly integrate out (read: leave out) the degrees of freedom of the outside world, thereby reducing the quantum mechanical description to the space spanned by the degrees of free-

49 dom of the studied system itself. Note: sometimes it is not really necessary to integrate out certain degrees of freedom, but it simply happens to be more convenient to do so. An example is the polarization of a particle beam, for which it is convenient not to have to deal with the spatial degrees of freedom (see 2.2). § Integrating out degrees of freedom: to get an idea of what happens when integrating out quantum mechanical degrees of freedom, we consider a spin-1/2 system described by the following normalized pure state function in the position representation:

ψ+(~r ) χ(~r ) = = ψ+(~r ) χ 1 , 1 + ψ (~r ) χ 1 , 1 , ψ (~r ) 2 2 − 2 − 2 − !

′ ′ with d~rψλ∗(~r )ψλ (~r ) = Wλ δλλ (λ,λ′ = ) and W+ + W = 1 . ± − Z For convenience we have taken ψ (~r ) to be orthogonal. Next we integrate out the spatial ± degrees of freedom in order to reduce the quantum mechanical description to the two- dimensional spin space. Consider to this end an arbitrary spin observable Aˆ, which acts as a 2 2 matrix A in spin space and which has no action in position space. The corresponding × expectation value for the state χ(~r ) then reads

Aˆ = d~rχ†(~r )A χ(~r ) = Tr A d~rχ(~r )χ†(~r ) Tr(Aρ) , h i ≡ Z Z ~r integrated out W 0 with ρ = d~r ~r χ χ ~r = + = density matrix in spin space z h }|| ih | {i 0 W Z − !

and Tr = trace in spin space .

Here we used that A commutes with ~r and we exploited the fact that for D-dimensional vectors v and D D matrices M the following holds: × D D D v∗M v = M (v v∗) M W = Tr(MW ) . i ij j ij j i ≡ ij ji i,j =1 i,j =1 i,j =1 X X X The expectation value for the complete system splits up according to

ˆ 1 1 † 1 1 † ˆ ˆ A = W+ Tr A χ , χ 1 , 1 + W Tr A χ , χ 1 , 1 = W+ A + + W A h i 2 2 2 2 − 2 − 2 2 − 2 h i − h i− into two independent expectation values, one for the pure spin state χ 1 1 with weight 2 , 2 W+ 0 and one for the pure spin state χ 1 , 1 with weight W = 1 W+ 0. The ≥ 2 − 2 − − ≥

50 quantum mechanical description in spin space is now determined by the density matrix ρ, which is composed of statistical weights and projection operators on pure states:

1 0 0 0 1 1 † 1 1 † ρ = W+ χ , χ 1 , 1 + W χ , χ 1 , 1 = W+ + W . 2 2 2 2 − 2 − 2 2 − 2 0 0 ! − 0 1 !

Mixed ensembles: integrating out quantum mechanical degrees of freedom gives rise to so-called mixed ensembles. These ensembles are incoherent mixtures (i.e. statistical mixtures) of pure subensembles, implying double statistics:

on the one hand the pure quantum mechanical state functions that describe the • pure subensembles have a statistical interpretation;

on the other hand the complete mixed ensemble comprises of a statistical mixture • of such subensembles.

The quantum physics in such scenarios is determined by the density operator, which contains the relevant properties of the mixed ensemble. These relevant properties are the pure quantum mechanical state functions that describe the pure subensembles and the corresponding statistical weights.

You might be inclined to think that a mixed ensemble somehow corresponds to a kind of superposition of pure state functions. As will become clear from the example given below, this is not the case.

Mixed ensembles vs pure ensembles: consider the previous example with equal statistical weights W =0.5, resulting in a statistical mixture with density matrix ρ = 1 I for ± 2 which 50% of the particles are in the spin state χ 1 , 1 and 50% in the spin state χ 1 , 1 . 2 2 2 − 2 Such an incoherent mixture of pure spin states χ 1 , 1 cannot be represented as a linear 2 ± 2 combination of states such as (χ 1 , 1 + χ 1 , 1 )/√2, in spite of the fact that such a pure 2 2 2 − 2 state indeed corresponds to a 50% chance to measure the particles in each of the two spin states. However, other measurable quantities exist that will result in completely diﬀerent

measurement results in both situations. For instance, the state (χ 1 , 1 + χ 1 , 1 )/√2 is a 2 2 2 − 2 pure eigenstate of the spin operator Sˆx belonging to the eigenvalue ~/2. Upon measuring the spin in the x-direction the outcome of the measurement will be ﬁxed: 100% of the particles will correspond to the measurement value ~/2. For the mixed ensemble, however, we have 1 ~ ρˆ = 2 1ˆ ~ ~ Sˆ = Tr(σ ρ) ======Tr(σ ) = 0 = , h xi 2 x 4 x 6 2 since none of the two types of particles on which the measurement is performed actually is in an eigenstate of Sˆx . An incoherent mixture in fact implies that the phases of

51 the separate components in the mixture are unrelated, which excludes a linear combination of pure state functions. The ensemble splits up in independent subensembles, each with a speciﬁc spin state and statistical weight.

Density operators ﬁnd applications in many branches of physics. In this chapter we will encounter a few characteristic examples from scattering theory and statistical physics (with applications in solid-state physics, nuclear physics, astrophysics and low-temperature physics). Density operators are also the tools of choice for the description of the state- function collapse of a microscopic system as caused by the inﬂuence of a macroscopic piece of measurement equipment.

2.1 The density operator (J. von Neumann, 1927)

Consider an ensemble consisting of a statistical mixture of N independent subensembles, each described by a normalized pure state function

ψ(α) α , with α α =1 (α =1, , N ) . (129) | i ≡ | i h | i ··· Each subensemble is a collection of identical, independent, identically prepared systems described by the corresponding state function α . The independent systems represented | i by the various state functions α are of identical composition, but they diﬀer by the state | i they are in. There is no need for these states to be eigenstates of one and the same observable, so that in general α α′ = 0 if α = α′. An example could be an electron beam of h | i 6 6 which 40% of the particles are polarized along the z-axis, 30% along the y-axis and 30% along the x-axis. This beam is composed of three independent subensembles that are each described by a pure spin function. The three types of systems described by these pure spin functions all consist of one electron, but the spin is quantized in a diﬀerent direction in each of the three cases.

From expectation value to ensemble average: take n to be an orthonormal set {| i} of eigenstates belonging to a complete set of commuting observables of the considered type of system. For discrete values of n we then have

n n′ = δ ′ and n n = 1ˆ . (130) h | i nn | ih | n X For n-values belonging to the continuous part of the spectrum, as usual δnn′ should be

replaced by δ(n n′) and the sum by an integral. Next we can decompose the normalized − pure state functions α in terms of this basis according to | i (130) α ==== n n α c(α) n , with α α = c(α) 2 = 1 . (131) | i | ih | i ≡ n | i h | i | n | n n n X X X

52 Let Aˆ be an observable pertaining to the considered type of system, with corresponding (classical) dynamical variable . For each of the normalized states α the expectation A | i value (= quantum mechanical average) of the dynamical variable is given by A ˆ ˆ (130) ˆ (131) (α) (α) ˆ A α α A α ==== α n′ n′ A n n α ==== cn cn′ ∗ n′ A n . (132) h i ≡ h | | i ′ h | ih | | ih | i ′ h | | i Xn,n Xn,n Subsequently we take into account that we are dealing with a statistical mixture of pure subensembles. To this end we indicate the statistical weight of the pure state α in the | i ensemble by W , corresponding to the fraction of systems in the state α . In short, the α | i factor W is the probability for the system to be in the state α . The ensemble average α | i (= statistical average) [Aˆ] of the dynamical variable can be expressed in a representa- A tion independent way as

N N [Aˆ] = W Aˆ , with W [0, 1] and W = 1 . (133) α h iα α ∈ α α=1 α=1 X X

Deﬁnition of the density operator: by means of equation (132) the expression for the ensemble average can be rewritten as

N [Aˆ] = W n α α n′ n′ Aˆ n n ρˆ n′ n′ Aˆ n , (134) α h | ih | ih | | i ≡ h | | ih | | i α=1 ′ ′ X Xn,n Xn,n where the density operator ρˆ of the ensemble is composed of projection operators on the pure states: N ρˆ = W α α . (135) α | ih | α=1 X Since W IR, this density operator is hermitian and therefore α ∈ N N (135) ψ ρˆ ψ ∗ ==== W ψ α ∗ α ψ ∗ = W ψ α α ψ = ψ ρˆ ψ . ∀ h 2| | 1i α h 2| i h | 1i α h 1| ih | 2i h 1| | 2i ψ1,ψ2 α=1 α=1 X X In the n-representation the density operator is characterized by the so-called density matrix

N (131),(135) (α) (α) ρ ′ = n ρˆ n′ ======W c c ′ ∗ . (136) nn h | | i α n n α=1 X In matrix language the ensemble average of the dynamical variable simply reads A

(134) (130) [Aˆ] ==== n ρˆ n′ n′ Aˆ n ==== n ρˆAˆ n = Tr(ˆρAˆ) = [Aˆ] . (137) h | | ih | | i h | | i ′ n Xn,n X

53 Properties of the density operator:

Since the density operatorρ ˆ is hermitian, an orthonormal basis exists of density- • operator eigenstates k with real eigenvalues ρ . Such an eigenvalue {| i} { k} N (136) ρ = k ρˆ k ==== W c(α) 2 [0, 1] (138) k h | | i α | k | ∈ α=1 X can be interpreted as the probability for the system to be in the pure state k . | i Conservation of probability implies • Tr(ˆρ)=1 . (139)

N N (137) (133) α α =1 (133) Proof: Tr(ˆρ) ==== [1]ˆ ==== W 1ˆ h====| i W ==== 1 . α h iα α α=1 α=1 X X For arbitrary ensembles we have • Tr(ˆρ2) 1 . (140) ≤ Proof: in terms of the eigenvalues ρ [0, 1] ofρ ˆ one readily ﬁnds k ∈ (137) (139) [ˆρ] ==== Tr(ˆρ2) = ρ2 ρ = Tr(ˆρ) ==== 1 . k ≤ k Xk Xk The following criterion holds for a system in a pure state: •

Tr(ˆρ2)=1 ρˆ2 =ρ ˆ the system is in a pure state . (141) ⇔ ⇔

Proof ( ): assume that the system is in the pure state λ . In that case Wα = δαλ ⇐ (135) | i and the density operatorρ ˆ ==== λ λ is a projection operator on λ . This | ih | | i automatically implies thatρ ˆ2 =ρ ˆ and Tr(ˆρ2) = Tr(ˆρ) = 1. Proof ( ): let Tr(ˆρ2) = Tr(ˆρ) = 1. In terms of the orthonormal basis k of ⇒ {| i} eigenstates ofρ ˆ we then have

ρ2 ρ 2 k ≤ k ρk = ρk = 1 === ρk = δkλ ⇒λ ∃ Xk Xk spectral decomposition :ρ ˆ = ρ k k = λ λ =ρ ˆ2 . ⇒ k | ih | | ih | Xk Because the density operator can apparently be expressed as a projection operator ρˆ = λ λ , it follows automatically that we are dealing with a pure ensemble with | ih | the system being in the pure state λ . | i

54 If the system happens to be in a pure state λ , then | i ρˆ = λ λ ρˆ (142) | ih | ≡ λ and all eigenvalues ofρ ˆ equal 0 except for one eigenvalue 1 belonging to the eigenstate λ . In this situation the ensemble average of the dynamical variable obviously | i A coincides with the expectation value for the state λ : | i

(133) [Aˆ] = Tr(ˆρ Aˆ) ==== Aˆ = λ Aˆ λ . (143) λ h iλ h | | i

2.2 Example: polarization of a spin-1/2 ensemble

As a ﬁrst application of the density-matrix formalism we consider a beam of spin-1/2 particles. A good understanding of the properties of such a beam is of particular interest for scattering experiments in high-energy physics, since at least one of the beams usually involves spin-1/2 particles (such as electrons, positrons, protons or antiprotons). In scattering experiments the beams are suﬃciently dilute to treat the beam particles as independent. In this sense the beam should not be regarded as an identical-particle system anymore, but rather as a (mixed) ensemble of beam particles that can be used to perform a repeated quantum mechanical scattering experiment.

The pure states of the beam particles can for instance be decomposed in terms of the basis ~p, m : ~p IR3 and m = 1/2 , where ~p is the momentum eigenvalue of {| si ∈ s ± } the beam particle and ms~ the spin component along the z-axis. Next we leave out the momentum variables from the quantum mechanical description and choose to work with the reduced density matrix in the 2-dimensional spin space. Spin information is actually an important tool to analyse scattering data in detail.

Preparatory step: combining two pure beams.

Consider two beams. Beam a consists of Na spin-1/2 particles prepared in the pure spin state χ(a) and beam b consists of N spin-1/2 particles prepared in the pure spin state | i b χ(b) . Subsequently both beams are merged. This results in a statistical mixture of the | i two pure subensembles with weights Wa = Na/(Na +Nb) and Wb = Nb/(Na +Nb). The density operator of this mixed ensemble is given by

ρˆ = W χ(a) χ(a) + W χ(b) χ(b) . (144) a | ih | b | ih |

With respect to the standard basis χ1 χ 1 , 1 , χ2 χ 1 , 1 we have {| i ≡ 2 2 | i ≡ 2 − 2 } χ(a) = c(a) χ + c(a) χ and χ(b) = c(b) χ + c(b) χ , (145) | i 1 | 1i 2 | 2i | i 1 | 1i 2 | 2i

55 giving rise to the density matrix

(a) (b) (a) (a) (b) (b) 2 2 ∗ ∗ (136) Wa c1 + Wb c1 Wa c1 c2 + Wb c1 c2 ρ ==== | | | | (146)  (a) (a) (b) (b) (a) 2 (b) 2  W c ∗c + W c ∗c W c + W c a 1 2 b 1 2 a | 2 | b | 2 |   in spin space. If the original pure beam states happen to coincide with the chosen basis states of spin space, i.e. χ(a) = χ and χ(b) = χ , then the density matrix simpliﬁes | i | 1i | i | 2i to a diagonal matrix:

Na W 0 0 a Na +Nb ρ = =  N  . (147) 0 Wb ! 0 b  Na +Nb    Polarization of an arbitrary spin-1/2 ensemble: consider the 2 2 density matrix ρ × in spin space. This matrix can be decomposed as

ρ = A I + A~ ~σ (A C and A C) , (148) 0 2 · 0 ∈ x,y,z ∈

in terms of the 2-dimensional identity matrix I2 and Pauli spin matrices σx , σy and σz that together form a basis for 2 2 matrices (see App. B). From the trace identities (B.5) × for these basis matrices we can deduce that

(B.5) (139) Tr(ρ) = A Tr(I ) + A~ Tr(~σ) ==== 2A ==== 1 , 0 2 · 0

(B.5) [σj] = Tr(ρσj ) = A0 Tr(σj) + Ak Tr(σkσj) ==== 2Ak δjk = 2Aj (j,k = x, y, z ) . Xk Xk ˆ In terms of the spin operator S~ = ~~σ/2 the density matrix is given by

1+Pz Px iPy 1 ~ 1 − ~ 2 ~ˆ 3 ρ = I2 + P ~σ = , P = [~σ ] = ~ [S ] IR . (149) 2 · 2  P + iP 1 P  ∈ x y − z  

The density matrix is determined by three (real) parameters Px,y,z that together form the polarization vector P~ of the ensemble. The fact that ρ has three degrees of freedom is relatively easy to comprehend. A complex 2 2 matrix that is hermitian and has a unit × trace is ﬁxed by 2 (2 2) 1 1 = 3 independent real parameters. × × × 2 − The physical interpretation of the polarization vector P~ : to this end we make use of the fact that the eigenvalues λ of the matrix 2ρ should satisfy

(1+ P λ)(1 P λ) P 2 P 2 = (1 λ)2 P~ 2 = 0 λ = 1 P~ . (150) z − − z − − x − y − − ⇒ ±| |

56 Hence, in diagonal form the density matrix looks as follows:

1 1+ P~ 0 ρdiag = | | , (151) 2 0 1 P~ −| | ! which follows from equation (149) by choosing the spin quantization axis (z-axis) parallel to P~ . Next deﬁne the spin eigenvectors χ and χ for spin quantization along P~ , so that ↑ ↓ (P~ ~σ)χ = P~ χ and (P~ ~σ)χ = P~ χ . Then it follows from equation (147) that · ↑ | | ↑ · ↓ −| | ↓ 1 N 1 N (1 + P~ ) = W = ↑ and (1 P~ ) = W = ↓ 2 | | ↑ N + N 2 −| | ↓ N + N ↑ ↓ ↑ ↓ N N P~ = W W = ↑ − ↓ [0, 1] , (152) ⇒ | | ↑ − ↓ N + N ∈ ↑ ↓ where N and N count the number of spin-1/2 particles in the spin states χ and χ . ↑ ↓ ↑ ↓ In that way P~ can be readily interpreted as the degree of polarization of the ensemble | | and ~e = P/~ P~ as the direction of polarization. P | | Special cases:

If the ensemble refers to a pure spin state, then we know because of equation (141) • that ρ = ρ2. From this it follows directly that ρ is now determined by two real parameters and a sign (as a result of squaring): by means of equation (149) as well as the relation

(B.6) (~σ A~ )(~σ B~ ) ==== (A~ B~ )I + i~σ (A~ B~ ) (A,~ B~ IR3 ) (153) · · · 2 · × ∈ the condition ρ = ρ2 amounts to

1 1 2 1 1+ P~ 2 ρ = I +P~ ~σ = ρ2 = I +P~ ~σ = I + P~ ~σ P~ = 1 . 2 2 · 4 2 · 2 2 2 · ⇒ | | In diagonal form the density matrix is given by the projection operator

diag 1 0 ~ ρpure = : 100% polarization with quantization axis parallel to P. 0 0 ! (154) This corresponds to maximal order (read: maximal quantum information), with all particles polarized in the direction of the polarization vector P~ .

For 0 < P~ < 1 the ensemble is partially polarized, represented by the inequality • | | 1 2 1 ~ 2 2 < Tr(ρ ) = 2 (1 + P ) < 1.

57 We speak of an unpolarized ensemble if P~ = 0. In that case • | | ρ = 1 I (155) P~ =~0 2 2

2 2 1 and Tr(ρ ) takes on its minimal value Tr(ρ )= 2 . In this situation as many particles are in the spin “ ” state as in the spin “ ” state. As such we are dealing with ↑ ↓ an equal admixture of two totally polarized subensembles, one with spin parallel to the quantization axis and one with spin antiparallel to the quantization axis. Note, though, that the direction of this quantization axis can be chosen freely! An unpolarized ensemble is in fact an example of a completely random ensemble with maximal disorder (see 2.4). § An ensemble (beam) with degree of polarization P~ can be viewed as being com- • | | posed of a totally polarized part and an unpolarized part. In diagonal form this reads (151) 1 0 1 P~ 1 0 ρdiag ==== P~ + −| | . | | 0 0 ! 2 0 1 !

Freedom to choose: the density matrix (155) of a completely random ensemble does not depend on the choice of representation for the considered space, in contrast to the density matrix (154) for a pure ensemble. Moreover, a given mixed ensemble can be decomposed in diﬀerent ways in terms of pure ensembles. For instance, a mixture with 20% of the particles polarized in the positive x-direction, 20% in the negative x-direction, 30% in the positive z-direction and 30% in the negative z-direction results in a net unpolarized en- 1 semble, since 0.2(ρ + ρ )+0.3(ρ + ρ ) = I2 . P~ =~ex P~ = −~ex P~ =~ez P~ = −~ez 2

2.3 The equation of motion for the density operator

As a next step towards quantum statistics we determine the equation of motion for the density operator in the Schrödinger picture. Consider to this end a statistical mixture of pure states that is characterized at t = t0 by the density operator N ρˆ(t ) = W α(t ) α(t ) . (156) 0 α | 0 ih 0 | α=1 X Assume the weights Wα of the statistical mixture not to change over time. Then the density operator evolves according to N α(t) = Uˆ(t, t ) α(t ) ρˆ(t) = W α(t) α(t) = Uˆ(t, t )ˆρ(t )Uˆ †(t, t ) . (157) | i 0 | 0 i ⇒ α | ih | 0 0 0 α=1 X As shown in the lecture course Quantum Mechanics 2, the evolution operator Uˆ(t, t0) satisfies the differential equation ∂ i~ Uˆ(t, t ) = Hˆ (t)Uˆ(t, t ) , (158) ∂t 0 0 58 where Hˆ (t) is the Hamilton operator belonging to the type of system described by the ensemble. From this we obtain the following equation of motion for the density operator:

d i~ ρˆ(t) = Hˆ (t), ρˆ(t) , (159) dt which is better known as the Liouville equation. This is the quantum mechanical analogue of the equation of motion for the phase-space probability density in classical statistical mechanics, which can be formulated in terms of Poisson brackets as ∂ρ /∂t = ρ , . cl −{ cl Hcl} For this reason the name “density operator” was given toρ ˆ.

Note:ρ ˆ(t) does not possess the typical time evolution that we would expect for a quantum mechanical operator. Sinceρ ˆ(t) is deﬁned in terms of state functions, it is time independent in the Heisenberg picture and time dependent in the Schr¨odinger picture. This is precisely the opposite of the behaviour of a normal quantum mechanical operator.

Time evolution of the ensemble average of the dynamical variable : the en- A semble average [Aˆ] as deﬁned in 2.1 satisﬁes the evolution equation §

d (137) d ∂Aˆ dˆρ [Aˆ ] ==== Tr(ˆρAˆ) = Tr ρˆ + Tr Aˆ dt dt ∂t dt (159) ∂Aˆ i ∂Aˆ i ==== Tr ρˆ Tr Hˆ ρÂˆ ρˆHˆ Aˆ = Tr ρˆ Tr ρÂˆHˆ ρˆHˆ Aˆ ∂t − ~ − ∂t − ~ − (137) ∂Aˆ i d ==== [AˆHˆ Hˆ Aˆ] = [Aˆ ] , (160) ∂t − ~ − dt where ∂A/∂tˆ refers to the explicit time dependence of Aˆ. In the penultimate step we have used that the trace is invariant under cyclic permutations, i.e. Tr(AˆBˆCˆ) = Tr(CÂˆBˆ) = Tr(BˆCÂˆ). The evolution equation (160) has the same form as the evolution equation for the expectation value of a dynamical variable in ordinary QM. However, this time we have averaged twice in view of the double statistics!

2.4 Quantum mechanical ensembles in thermal equilibrium

As we have seen, there are marked diﬀerences between pure ensembles (with maximal order) and completely random ensembles (with maximale disorder). We are now going to investigate the generic diﬀerences a bit closer.

Let k be an orthonormal set of eigenstates ofρ ˆ corresponding to the eigenvalues ρ . {| i} { k} As such, these eigenstates together span the (reduced) D-dimensional space on which the pure state functions of the subensembles are deﬁned. This dimensionality D indicates the maximum number of independent quantum states that can be identiﬁed in the reduced

59 space on which we have chosen to consider the quantum mechanical systems. For instance, the density matrix in the spin-1/2 spin space has dimensionality D = 2. With respect to this basis we have

0 . .. Ø 1   0 1 1 Ø  ..  ρpure =  1  vs ρrandom = . .  0  D Ø 1  .     Ø ..   1       0        Examples of these extreme forms of density matrices have been given in equations (154) and (155) in 2.2. In case of a pure ensemble the density matrix is simply the projection § matrix on the corresponding pure state vector, which depends crucially on the chosen representation of the D-dimensional space. In case of a completely random ensemble, the D orthonormal basis states each receive the same statistical weight 1/D to guarantee that Tr(ˆρ) = 1. In that case each state is equally probable and the density matrix is proportional to the D-dimensional identity matrix, which does not depend at all on the chosen representation.

In order to quantify the diﬀerences we introduce the quantity

D D compl. S σ Tr(ˆρ lnρ ˆ) ==== k ρˆ k′ k′ lnρ ˆ k = ρk ln(ρk) , (161) ≡ − − ′ h | | ih | | i − ≡ kB k,kX=1 kX=1 where ρ represents the probability to ﬁnd the system in the pure basis state k . This k | i quantity is minimal for pure states and maximal for a completely random ensemble:

D 1 σ = 0 vs σ = ln(1/D) = ln(D) . (162) pure random − D kX=1 That σ is maximal will be proven in 2.4.3. In accordance with classical thermo- random § dynamics, it can be deduced from σ = S/kB and σrandom = ln(D) that the quantity 4 S should be interpreted as the quantum mechanical entropy and kB as the well-known Boltzmann constant. You could even say that this deﬁnition of the entropy is superior to the classical one. In classical mechanics there is no such thing as counting states. At best one could work with phase-space volumes that have to be made dimensionless

4This deﬁnition of entropy is also used in information theory in the form of the so-called Shannon

entropy Pk ln(Pk), which is an inverse measure for the amount of information that is encoded in − k the probability distribution P1, , PN . Each type of probability distribution in fact corresponds to a P { ··· } speciﬁc type of ensemble. In the lecture course Statistical Mechanics a relation will be established between the thermodynamical and quantum mechanical deﬁnitions of the entropy of a canonical ensemble.

60 by means of an arbitrary normalization factor. For this reason the classical entropy can only be deﬁned up to an additive constant. In QM, however, there is a natural unit of counting/normalization in the form of ~.

2.4.1 Thermal equilibrium (thermodynamic postulate)

A quantum mechanical ensemble is regarded as being in thermal equilibrium if it eﬀectively satisﬁes the equilibrium conditions

∂ d (159) Hˆ = 0 and ρˆ = 0 === H,ˆ ρˆ =0 (163) ∂t dt ⇒ and if the corresponding constant entropy is maximal. This last condition on the entropy is the quantum mechanical analogue of the classical second law of thermodynamics, which states that the entropy of macroscopic systems in non-equilibrium conﬁgurations always increases until equilibrium is reached.

In thermal equilibrium all averaged system quantities [Aˆ] are constant, provided that ∂A/∂tˆ = 0. For example this holds for the average energy of the system, i.e. Aˆ = Hˆ .

d (160), ∂A/∂tˆ = 0 dˆρ (163) Proof : [Aˆ] ======Tr Aˆ ==== 0 . dt dt In thermal equilibrium the density operator commutes with the Hamilton operator. There- fore an orthonormal set of simultaneous eigenfunctions k of both Hˆ andρ ˆ exists: {| i} Hˆ k = E k andρ ˆ k = ρ k . (164) | i k | i | i k | i Using this basis we will maximize the entropy in three diﬀerent scenarios in 2.4.2 – 2.4.4. §

2.4.2 Canonical ensembles (J.W. Gibbs, 1902)

heat bath

NR ,VR , T

system N,V,T ∆E

(N,V,T)-ensemble closed bottle in a water container

61 Consider a system with a constant number N of identical particles inside a macroscopic ﬁnite enclosure with ﬁxed volume V . Assume the considered system to be in thermal equilibrium with a very large heat bath (outside world), with which it is in weak energy contact. Together the heat bath and the embedded system form a closed system (isolated system).

Weak energy contact: by weak energy contact we mean that the embedded system and heat bath can exchange energy, since only the total energy of the entire isolated system is conserved, but that they do so without having a noticeable influence on each other’s energy spectra. Since the mutual interaction is that weak, both the heat bath and the embedded system can be in a definite energy eigenstate at every instance of time (separation of variables). The system and heat bath merely depend on each other through the weak energy contact, which makes the energy of the system time dependent. This weak energy contact does have a role to play, since it provides the mechanism by which the system can reach thermal equilibrium with the heat bath. A weak energy contact can for instance be achieved by separating the system from the heat bath by means of a macroscopic partition wall. This allows energy to be transferred by means of collisions with the atoms in the wall. In that case only a negligible fraction of the enclosed particles will feel the presence of the partition wall if the average de Broglie wavelength of the particles is much smaller than the dimensions of the enclosure (see 2.5). For instance this applies to systems with § sufficiently large (macroscopic) enclosures and/or sufficiently high temperatures.

The heat bath determines the mixture: we will see that in thermal equilibrium all energy eigenstates of a closed system have equal probability, i.e. a closed system is described by a completely random ensemble. For a fixed total energy of the entire closed system the number of energy eigenstates of the heat bath is largest if the energy of the heat bath is highest and therefore the energy of the embedded N-particle system is lowest. Subsequently the degrees of freedom of the large heat bath are integrated out, which causes the embedded N-particle systems with the lowest energies to automatically have the highest statistical weights. This gives rise to a so-called canonical ensemble, which is sometimes referred to as (N,V,T )-ensemble. This is a statistical mixture of N-particle systems in specific energy eigenstates. As a result of the weak energy contact with the heat bath, such an N-particle system in thermal equilibrium will not have a fixed energy value. However, the ensemble average E¯ will remain constant in thermal equilibrium.

The canonical density operator: in order to determine the density operator of a canonical ensemble in thermal equilibrium, we have to maximize the entropy under the additional constraints Tr(ˆρ)=1 and Tr(ˆρHˆ ) = [Hˆ ] E¯ = constant. To this end we employ the ≡ Lagrange-multiplier method (see App. C) by varying with respect to the eigenvalues ρk as

62 well as the Lagrange multipliers, i.e.

(164) 0 = δ σ β Tr(ˆρHˆ ) E¯ λ Tr(ˆρ) 1 ==== δ βE¯+λ ρ ln(ρ )+βE +λ − − − − − k k k k X for all variations δρk , δβ and δλ. Note: there is no need to vary with respect to the

energy eigenvalues Ek , as these energy eigenvalues are independent of the statistical mixture in case of a weak energy contact. From the Lagrange-multiplier method it follows automatically that the constraint conditions have to be satisﬁed as well as the condition

δρ ln(ρ )+1+ βE + λ = 0 ρ = exp( 1 λ) exp( βE ) . ∀ k k k ⇒ k − − − k δρk k X The Lagrange multiplier λ can now be eliminated by means of the constraint condition

Tr(ˆρ) = exp( 1 λ) exp( βEk′ ) =1 , − − ′ − Xk which implies for ρk that exp( βEk) ρk = − . (165) exp( βEk′ ) k′ − In these sums the summation runs overP all fully speciﬁed N-particle energy eigenvalues, including the complete degree of degeneracy of these energy levels. Subsequently we use the Lagrange multiplier β to deﬁne the temperature of the considered ensemble:

1 T . (166) ≡ kBβ

Out of the complete disorder of the closed system, the contact with the heat bath has created order in the energy distribution of the embedded N-particle system. Note: if the N-particle ground state is not degenerate, then the canonical ensemble will change in the low-temperature limit T 0 (β ) into a pure → →∞ ensemble where all N-particle systems will be in the ground state.

Finally we introduce the canonical partition function (normalization factor)

Z (T ) exp( βE ′ ) = k′ exp( βHˆ ) k′ = Tr exp( βHˆ ) . (167) N ≡ − k h | − | i − k′ k′ X X In spectral decomposition the density operator of a canonical ensemble then reads

(164) (165) exp( βEk) 1 ρˆ ==== ρk k k ==== − k k = exp( βHˆ ) k k | ih | ZN (T ) | ih | ZN (T ) − | ih | Xk Xk Xk

compl. 1 exp( βHˆ ) === ρˆ(Hˆ ) = exp( βHˆ ) = − . (168) ⇒ ZN (T ) − Tr exp( βHˆ ) − 63 The label N in ZN (T ) refers to the number of particles of the considered type of embedded system. An arbitrary averaged physical quantity of the embedded system is then given by the ensemble average 1 [Aˆ] = Tr Aˆ exp( βHˆ ) , (169) ZN (T ) − allowing us to derive the average system energy from the partition function:

Tr Hˆ exp( βHˆ ) ¯ ˆ ∂ZN (T )/∂β ∂ E = [H ] = − = − = ln ZN (T ) . (170) ZN (T ) ZN (T ) − ∂β Number of particles and the canonical-ensemble concept: the embedded systems of a canonical ensemble can just as well consist of a single particle as of a macroscopic number of particles (such as a gas). A popular way to link up with the classical thermodynamics of ideal gases is to single out one particle in a larger gas system and consider the corresponding (single-particle) canonical ensemble. This approach of treating the rest of the gas as part of the heat bath has its limitations, though. The canonical-ensemble concept implies that the influence exerted by the particles of the heat bath on the particle of the embedded system is exclusively limited to a weak energy contact. This obviously breaks down if the particle density is too high, causing the quantum mechanical overlap between the identical particles in embedded system and heat bath to become relevant. In that case also the effective quantum mechanical interaction induced by the (anti)symmetrization procedure should be taken into account. These quantum mechanical effects will become more pronounced with increasing particle densities (see 2.5 – 2.7). In the two examples given below we will § nevertheless consider two canonical ensembles of systems that consist of a single particle. The reason to do so is to first link up with classical statistical physics before zooming in on the quantum mechanical aspects. In order to study these many-particle aspects, the grand-canonical ensemble approach will turn out to be more handy.

Equipartition of energy: as a standard example we consider a canonical ensemble of systems that each consist of a single free spin-0 particle with mass m that is contained in a macroscopic enclosure (box). Since quantum mechanical many-particle aspects evidently do not play a role in a 1-particle system, we expect the average energy per particle in that case to obey the classical principle of equipartition of energy for an ideal gas. This states that each kinetic and elastic degree of freedom of the considered type of system (i.e. a position/momentum degree of freedom that contributes quadratically to the Hamiltonian)

will contribute kBT/2 to the average energy. This is indeed conﬁrmed by the quantum mechanical calculation (see exercise 13), which tells us that the quantum mechanical deﬁ- nitions of entropy, temperature and Boltzmann constant are consistent with their classical thermodynamic counterparts.

64 Example: magnetization. Consider a canonical ensemble of systems that each consist of a single free electron that is experiencing a constant homogeneous magnetic ﬁeld in the z-direction, ~ = ~e . In spin space this corresponds to an interaction B B z

spin ˆ 2µB Hˆ = ~ S ~ = B Sˆz , (171) B − M · B ~

ˆ ˆ spatial ˆ spin in terms of the Bohr magneton µB . The full Hamilton operator H = H + HB of the electron commutes with Sˆz and therefore the same should hold forρ ˆ(Hˆ ). Hence, the

density matrix is diagonal in spin space if we use a basis of eigenvectors of Sˆz to describe it. For a given spatial energy level the density operator has the following form in spin space: 1 exp( βµ ) 0 ρ spin = − B B . (172) exp( βµ ) + exp(βµ ) 0 exp(βµ ) − B B B B B B ! This ensemble corresponds to the following polarization vector:

spin cosh(βµB ) I2 sinh(βµB ) σz 1 ρ = B − B = I2 tanh(βµB ) σz 2 cosh(βµB ) 2 − B B (149) 2 === P~ = tanh(βµ )~e = [Sˆ ] ~e . (173) ⇒ − B B z ~ z z From this we can derive the magnetization, i.e. the average magnetic moment per electron in the direction of the magnetic ﬁeld:

1 ˆ 2µB M [ ~ ~ ] = [Sˆ ] = µ tanh(βµ ) . (174) S ≡ MS · B − ~ z B B B B If βµ 1, for instance when the magnetic ﬁeld is weak or the temperature high, then B B≪ we obtain Curie’s law:

2 2 (166) µB dMS µB 1 MS µB (βµB ) ==== χS (T ) = T − , (175) ≈ B kBT B ⇒ ≡ d =0 kBT ∝ B B

where χS (T ) is called the paramagnetic susceptibility of the spin ensemble.

2.4.3 Microcanonical ensembles

If we do not impose the constraint Tr(ˆρHˆ ) = E¯ in the previous derivation, then we will obtain the ensemble with maximum disorder once we maximize the entropy. After all, there is no heat bath to integrate out and create any order in the ensemble. We speak in that case of a microcanonical ensemble or (N,V,E)-ensemble, which determines the thermodynamical properties of a closed system in thermal equilibrium. Because of the absence of energy contact, the system energy is constant and not just the average value of

65 the system energy. If we take the density matrix ρ to have dimensionality D, then we ﬁnd by employing the Lagrange-multiplier method that

D (164) δ σ λ Tr(ˆρ) 1 ==== δ λ ρ ln(ρ )+ λ = 0 − − − k k k =1 X for all variations δρk and δλ. From this it follows that Tr(ˆρ) = 1 and ln(ρk)+1+ λ = 0. By combining the two conditions ρ = exp( 1 λ) and Tr(ˆρ) = D exp( 1 λ) = 1, k − − − − we arrive at the typical diagonal form ρ = 1/D as given in 2.4 for a completely k § random ensemble. This also proves the conjecture that the density matrix of a completely random ensemble gives rise to the absolute maximum of entropy. Note also that from equation (168) it can be read oﬀ that a microcanonical ensemble coincides with the high-temperature limit T (β 0) of a canonical ensemble. →∞ →

2.4.4 Grand canonical ensembles (J.W. Gibbs, 1902)

reservoir

α,VR , T

system α,V, T ∆E , ∆N

(α,V,T) or (µ,V,T)-ensemble open bottle in a water container

Finally, we consider a quantum mechanical ensemble that satisﬁes the same criteria as for a canonical ensemble with the exception that the embedded system is allowed to exchange both energy and particles with a reservoir. The (open) many-particle systems described by the ensemble have a variable number of particles. However, the total many-particle

Hamilton operator Hˆtot of such a many-particle system satisﬁes certain particle-number conservation laws.5 We know that the density operator has to be a constant of motion and that any order in the ensemble is imparted by the contact with the reservoir. As such, we expect the density operator to depend exclusively on observables that belong to conserved quantities that can be exchanged between reservoir and open system. In the considered case that would be the total many-particle Hamilton operator and the conserved

5For particles that are not constrained by conservation laws, such as thermal photons, only a canonical- ensemble approach is formally applicable. This will be explained in chapter 4 during the discussion of photon-gas systems.

66 combinations of total number operators. Let’s for convenience assume that the open system consists of one type of particle, i.e. we exclude the possibility of particle mixtures, and that no reactions can occur that affect the number of particles (such as particle decays). In that case Hˆtot has to be additive, i.e. Hˆtot, Nˆ = 0, to guarantee that Nˆ is a constant of motion. In this way the number of particles plays the same role as energy did in the canonical case, simply because the open system has no fixed number of particles but the complete closed system including the reservoir does. In thermal and diffusive equilibrium, the resulting type of ensemble is called a grand canonical ensemble,(α,V,T)-ensemble, or (µ,V,T)-ensemble.

Apart from the average energy we now also have to impose a constraint on the average number of particles of the open system while maximizing the entropy. For this purpose we use the total number operator Nˆ that counts the total number of particles in the open system, which has eigenvalues 0, 1, 2, . Apart from the Hamilton operator also ··· the density operator is additive, in order to guarantee that the entropy is additive. Hence, the total number operator commutes with both the Hamilton operator and the density operator. Bear in mind, though, that these two operators will depend on the precise number of particles present in the open system. This gives rise to the following set of mutually commuting observables N,ˆ Hˆtot(Nˆ) andρ ˆ(Nˆ), with corresponding orthonormal basis k, N : {| i} Nˆ k, N = N k, N , Hˆ (Nˆ) k, N = E (N) k, N | i | i tot | i k | i andρ ˆ(Nˆ) k, N = ρ (N) k, N . (176) | i k | i

The grand canonical density operator: in order to determine the density operator of a grand canonical ensemble in thermal and diffusive equilibrium we have to maximize the entropy under the additional constraints Tr(ˆρ) = 1, [Hˆ (Nˆ)] E¯ = constant tot ≡ tot and [Nˆ ] N¯ = constant. By means of the Lagrange-multiplier method we find ≡ 0 ==== δ σ α [Nˆ] N¯ β [Hˆ (Nˆ)] E¯ λ Tr(ˆρ) 1 − − − tot − tot − − (176) ==== δ αN¯ + βE¯ + λ ρ (N) ln ρ (N) + αN + βE (N)+ λ (177) tot − k k k N k X X for all variations δρk(N), δα, δβ and δλ. From this it follows automatically that the constraint conditions have to be satisfied as well as the condition

δρ (N) ln ρ (N) +1+ αN + βE (N)+ λ = 0 . ∀ k k k δρk(N) N k X X

67 Combined with the constraint Tr(ρ) = 1 this results in

exp βEk(N) αN ρk(N) = − − . (178) exp βEk′ (N ′) αN ′ N ′ k′ − − P P As in equation (166), the temperature T of the ensemble is deﬁned through T 1/k β . ≡ B Finally, we introduce the grand canonical partition function

(167) (α, T ) exp βE ′ (N ′) αN ′ ==== exp( αN ′) Z ′ (T ) (179) Z ≡ − k − − N N ′ k′ N ′ X X X and rewrite the grand-canonical density operator as

1 ρˆ = exp βHˆ (Nˆ) αNˆ =ρ ˆ Hˆ (Nˆ), Nˆ . (180) (α, T ) − tot − tot Z The quantity exp( α) can be regarded as the fugacity of the ensemble, which describes − how easy it is to move a particle from the reservoir to the open system:

if exp( α) is small, than contributions from small N values dominate and quantum • − mechanical many-particle aspects are not important;

if exp( α) is not small, than particle exchange is easier and quantum mechanical • − many-particle aspects are relevant.

Often the fugacity is written as exp(βµ), with µ = α/β known as the chemical potential. − The average total energy and the average number of particles of the open system can be extracted directly from the grand canonical partition function:

∂ ∂ E¯ = ln (α, T ) and N¯ = ln (α, T ) . (181) tot − ∂β Z − ∂α Z Remark: in scenarios that do either involve chemical reactions, particle decays or relativistic energies, we have to abandon conservation laws for individual types of particles. However, in those cases certain combinations of particle numbers often will be conserved. While maximizing the entropy, these generalized conservation laws will have to be imposed as constraints on the corresponding ensemble averages. As before this can be implemented by means of the Lagrange-multiplier method, assigning to each conservation law a spe- ciﬁc Lagrange multiplier and a related chemical potential. For instance, for relativistic energies particle–antiparticle pair creation can occur, so that only the diﬀerence between the number of particles and antiparticles remains conserved (which is equivalent to charge conservation). Another example is provided by a statistical mixture of A-, B- and C- particles for which the reversible reaction A ⇋ B + C is in chemical equilibrium. In that case the particle-number combination 2NA + NB + NC is conserved.

68 Possible realization: if we are dealing with a vary large number of particles inside a macroscopic volume, then the grand canonical ensemble can be realized straightforwardly by partitioning the macroscopic volume in very many identical cells. The inherent condition is that the cells themselves are again suﬃciently macroscopic to guarantee that the energy contact with the reservoir (read: the other cells) can be regarded as weak. Repeated measurements can be performed on such an ensemble by simply measuring locally. The internal structure of a star can, for instance, be determined along these lines by partitioning the star in macroscopic volume elements that are of negligible size compared to the dimensions of the star itself (see exercise 18). The relevant “measurable” quantities are in that case the local particle densities and the local total energy density.

2.4.5 Summary

The differences between the three types of ensembles reside in the increasing degree of order that is imparted on the embedded systems by the contact with the reservoir. In the microcanonical case this contact is extremely weak. It is that weak that we could effectively speak of the absence of any contact, causing E and N to be effectively fixed. Thermal equilibrium is in that case equivalent with the absence of any order. In the canonical case energy transfer occurs between reservoir and embedded system, with the total energy being conserved. Since the reservoir has so many more degrees of freedom than the embedded system, in thermal equilibrium order is generated in the energy mix of the embedded system with a preference for lower energies (i.e. higher reservoir energies). In the grand canonical case we add to this the possibility of particle exchange, with the corresponding particle conservation law translating into extra order in the particle mix of the embedded open system. The three different types of ensembles give rise to identical physical results if the relative

fluctuations around the equilibrium values E¯tot and N¯ are very small, causing the energy and number of particles of the embedded system to be effectively fixed. It is then a matter of choice which type of ensemble is more suitable for performing calculations (which in most cases will be the grand canonical ensemble). For instance this applies to systems consisting of a very large number of particles, in view of the extra factors 1/√N¯ that in general occur in the average relative fluctuations on the measurements.

2.5 Fermi gases at T = 0

In nature many physical systems can be modelled pretty well by means of an N-particle system consisting of a large number of non-interacting identical fermions inside a finite enclosure (box) with fixed volume. If the size of the box and the number of particles N are sufficiently large, then the precise shape of the box will not matter in the analysis. This type of many-particle system is called a Fermi gas. Examples are conduction electrons in a metal (see 2.5.3), heavy nuclei (see 2.5.4), compact imploded stars (see 2.5.5), etc. § § §

69 For determining the properties of such a Fermi gas z we consider a large cube with edges L and with the origin in one of the vertices (see picture). The volume of the cube is then given by V = L3. As a ﬁrst step we assume the cube to contain N free identical fermions with spin s and mass m. Scenarios where L y the particles experience a constant attractive poten- L tial or a constant homogeneous magnetic ﬁeld are x O considered in a next step. L

The 1-particle Hamilton operator for a free particle in a box is given by

~pˆ 2 0 inside the box Hˆ = + V (~rˆ) , V (~r ) = . (182) 2m  otherwise  ∞ This (kinetic) 1-particle Hamilton operator does not depend on the spin of the particle. Therefore, we start by solving the spatial eigenvalue equations and subsequently take into account the spin degeneracy. The spatial 1-particle energy eigenfunctions have to satisfy the position-space equation

~pˆ 2 ~2 2 ψ (~r ) = ~ ψ (~r ) = E ψ (~r ) , 2m ~ν − 2m ▽ ~ν ν ~ν

with ψ~ν(~r ) = 0 outside and on the edge of the box . (183)

For the cubic box the problem reduces to solving the well-known eigenvalue problem for a particle in a 1-dimensional inﬁnite potential well (a.k.a. inﬁnite square well). The normalized spatial energy eigenfunctions form standing waves inside the cube (see App. A.1):

2 3/2 ν π ν π ν π ψ (~r ) = sin x x sin y y sin z z , (184) ~ν L L L L ~2π2 ~2π2 ~2π2 E = (ν2 + ν2 + ν2) = ~ν 2 ν2 (ν =1, 2, ) . ν 2mL2 x y z 2mL2 ≡ 2mL2 x,y,z ···

Since E only depends on ν = ~ν and not on ν separately, almost all kinetic energy ν | | x,y,z eigenvalues are degenerate. Adding spin to the game ﬁnally leads to

ψ (~r, σ) = ψ (~r )χ (σ) (m = s, s+1, ,s 1,s) , (185) ~ν,ms ~ν s,ms s − − ··· −

where the 1-particle spinvector χs,ms (σ) as usual refers to a spin-s particle with spin

component ms~ along the z-axis. As a result, the degree of degeneracy of the energy

eigenvalues Eν is increased by a factor 2s+1.

70 νz

1 0 1 2 3 4 5 ν 1 y 2 3 4 5

νx

Next we make use of the fact that the box has macroscopic dimensions. As a result, the 1-particle energy levels are very closely spaced and it makes good sense to switch to a continuous density of states D(Ekin) that counts the number of 1-particle quantum states per unit of kinetic energy. In the continuum limit the number of 1-particle quantum states

with kinetic energy on the interval (Ekin , Ekin + dEkin) is simply given by D(Ekin)dEkin .

In order to determine D(Ekin) it will prove handy to introduce the quantized wave vector

~k = π~ν/L (ν =1, 2, ) , (186) x,y,z ··· to determine the corresponding density of states D(k), and ﬁnally to insert the expression 2 2 Ekin(k) = ~ k /(2m) for the kinetic energy in terms of the wave vector. As indicated in the picture given above, a unit cube in ~ν-space contains exactly one spatial quantum state and therefore 2s+1 fully speciﬁed quantum states (including spin). In view of the

positivity conditions νx,y,z > 0, all possible ~ν-values are in one octant of ~ν-space.

νy νy ν =4 ν = 11

νx νx

71 For ν-values that are not too small, the number of states N(k) with wave vector smaller than k = πν/L is in good approximation related to the corresponding volume in ~ν-space (as illustrated in the preceding picture for a 2-dimensional lattice):

k 1 4 3 (186) 2s +1 3 N(k) = dk′ D(k′) (2s + 1) πν ==== Vk . (187) ≈ 8 3 6π2 Z0 By taking a derivative we obtain

d 2s +1 D(k) N(k) = Vk2 . (188) ≈ dk 2π2

This expression is actually valid for an arbitrary spin-s particle in a box. Only

when we switch to D(Ekin) diﬀerences will occur between for instance relativistic and non-relativistic scenarios, since in those cases the dispersion relation

between Ekin and k will be diﬀerent (see exercise 17).

2 In the non-relativistic case considered here the dispersion relation reads k = 2mEkin/~ and therefore p

3/2 dk 2s +1 2m 1/2 D(Ekin) = D(k) = 2 2 V Ekin . (189) dEkin 4π ~ The approximation that we just used is obviously not correct for very low kinetic 1-particle energies. For instance, for k < π/L there are no kinetic energy levels possible at all. In fact, in the low-energy regime the discrete character of the energy levels manifests itself. This is caused by the fact that the box has ﬁnite dimensions that cannot be considered macroscopic for de Broglie wavelengths λ(k)=2π/k = (L). As we will see later, such low O kinetic energies play a relevant role in bosonic gases at very low temperatures (see 2.7). § They do not play an important role in a Fermi gas in view of the small number of diﬀerent (187) low-energy states N(k) ==== [V/λ3(k)] (see later). O

Geometrical aspects: the number of quantum states per unit of kinetic energy and per

unit of volume of the box is given in the continuum limit by D(Ekin)/V , which does not depend on the volume! In the continuum limit the precise shape of the box in general does not matter. If we indicate the surface area of the macroscopic box by S , then the

influence of the edge of the box on D(Ekin) is suppressed by a relative factor S/(kV ) (see exercise 15). This simply underlines the quantum mechanical expectation that the presence of the edge of the box is felt up to order λ(k) distances away. The corresponding suppression factor Sλ(k)/V is small if the kinetic energy is sufficiently high, as is true (on average) for a Fermi gas that has a sufficiently large number of particles.

72 Fermi gas in a constant attractive potential: as we will see in 2.5.3 and 2.5.4, § § in certain physical scenarios the Fermi gas is bound by a constant attractive potential V = V < 0 inside the box. This has no bearing on the kinetic energy levels nor on the − 0 corresponding energy eigenfunctions that were derived previously. However, the 1-particle energy levels will shift according to

V0 =0 E 6 E V , (190) ν −−−−→ ν − 0

where Eν is the kinetic contribution to the total 1-particle energy. By means of the deﬁnition E E V V (191) ≡ kin − 0 ≥ − 0 the 1-particle density of states of the bound system is expressible in terms of the 1-particle density of states (189) for free particles inside the box:

DV0 (E) = D(Ekin) = D(E + V0) . (192)

Fermi gas in a constant homogeneous magnetic ﬁeld: assume the Fermi gas to be subjected to a constant homogeneous magnetic ﬁeld in the z-direction, ~ = ~e . In spin B B z space this gives rise to an additional interaction

Hˆ spin = C Sˆ (C IR , C ) . (193) B z ∈ ∝ B Since this operator acts exclusively on spin space and the Hamilton operator (182) exclusively on position space, the 1-particle energy eigenfunctions are still given by equation (185). However, the (2s+1)-fold spin degeneracy of the kinetic 1-particle energy levels will be lifted according to

=0 E B 6 E + m ~C E , (194) ν −−−−→ ν s ≡ ν,ms where Eν is the kinetic contribution to the total 1-particle energy Eν,ms for a particle with spin component ms~ along the z-axis. For each value of the (magnetic) spin quantum number ms this is equivalent to a constant potential shift V = ms~C. By deﬁning

E E + m ~C m ~C, (195) ms ≡ kin s ≥ s the density of states for each of the 2s+1 individual spin states is expressible in terms of the 1-particle density of states (189) for free particles inside the box:

1 1 D (E ) = D(E ) = D(E m ~C ) . (196) ms ms 2s +1 kin 2s +1 ms − s

73 2.5.1 Ground state of a Fermi gas

In 2.4.2 it was shown that a gas at temperature T = 0 will be in the ground state § (completely degenerate gas). For gases consisting of bosons or distinguishable particles this automatically implies that all particles will populate the lowest 1-particle energy eigenstate. For a Fermi gas this is not a possibility, bearing in mind that Pauli’s exclusion principle tells us that at most one fermion can occupy a fully speciﬁed quantum state. In that case the N particles will occupy the kinetic 1-particle energy eigenstates one by one from the bottom up, with the highest occupied level given by the Fermi energy E ~2k2 /(2m). States with E > E will not be occupied (vacant) for a Fermi gas at F ≡ F kin F T = 0 and can only by excited by increasing the temperature or by including pair interactions into the analysis. The collective set of states with E E form the so-called kin ≤ F Fermi sea. The value of the Fermi energy can be extracted readily from equation (187), since

kF (187) 2s +1 N = N(k ) = dk D(k) ==== Vk3 (197) F 6π2 F Z0 implies that

6π2 1/3 ~2k2 ~2 6π2 2/3 k = ρ1/3 and E = F = ρ2/3 (198) F 2s +1 N F 2m 2m 2s +1 N

in terms of the particle density ρN = N/V . Since N is large, the precise degree of occupation of the (degenerate) highest energy level is immaterial. Moreover, the fact that only a limited amount of particles will occupy the lowest kinetic energy eigenstates also implies that it is a good approximation to replace the density of states by the continuous function in equation (189). Note that E ρ2/3/m, which tells us that the F ∝ N Fermi energy increases with increasing particle density and that it will intrinsically be a factor 103 smaller for protons and neutrons than for electrons. The total kinetic energy of the ground state of the Fermi gas amounts to

kF 2 2 2 ~ k (188) 2s +1 ~ (197),(198) 3 ET =0 = dk D(k) ==== Vk5 ======NE . (199) kin,tot 2m 10π2 2m F 5 F Z0 As a consequence, the average kinetic energy per particle is pretty substantial in spite of the extremely low temperature:

ET =0 3 E¯ = kin,tot = E . (200) kin N 5 F

74 Quite often a kind of eﬀective (classical) temperature is assigned to a T = 0 Fermi gas. This is done by invoking the classical principle of equipartition of energy:

T =0 2/3 Ekin,tot (199) 2 EF 2 ρN Tcl ==== TF , (201) ≡ 3NkB/2 5 kB ≡ 5 ∝ m

where TF is called the Fermi temperature. For a given constant number of particles such a completely degenerate Fermi gas actually exerts a ﬁnite pressure (degeneracy pressure)

T =0 T =0 5/3 ∂E (198),(199) 2 E (201) ρ P = kin,tot ======kin,tot ==== k ρ T N , (202) − ∂V 3 V B N cl ∝ m !N which obeys a kind of effective (classical) ideal gas law. The higher the particle density of the Fermi gas becomes, the more the particles lose their individuality and the higher the effective temperature will be, making it more difficult to excite the fermions. For finite temperatures T T the Fermi gas will actually behave as if we were dealing with a ≪ F completely degenerate Fermi gas (see 2.7). §

Calculational tool: to get a feel for the particle-speciﬁc aspects of the relation between rel EF , TF and ρN , a few characteristic values are listed in the table below. In this table ρN rel 2 and TF mark the particle density and Fermi temperature for which EF = mc , so that ρ ρrel and T T rel are required for the non-relativistic approach to be valid: N ≪ N F ≪ F

2 rel 2/3 rel rel 2/3 EF = mc ρN /ρN and TF = TF ρN /ρN . (203) 2 rel rel 3 particle mc in MeV TF in K ρN in m− electron 0.51100 0.5930 1010 1.6589 1036 × × proton 938.27 1.0888 1013 1.0269 1046 × × neutron 939.57 1.0903 1013 1.0312 1046 × ×

Ground state of a Fermi gas in a constant attractive potential: assume the Fermi gas to be bound by a constant attractive potential V = V < 0 inside the box. All quan- − 0 T =0 ¯ tities related to the kinetic part of the energy remain unaﬀected, such as EF , Ekin,tot , Ekin and the thermal gas pressure P . However, the total energy of the completely degenerate Fermi gas will undergo a shift: 3 3 ET =0 = NE NV E¯ V E¯ = V E , (204) tot 5 F − 0 ⇒ b ≡ 0 − kin 0 − 5 F

with E¯b the average binding energy per particle.

75 2.5.2 Fermi gas with periodic boundary conditions

In view of the applications in chapter 1 and the discussion of relativistic quantum theories, we make a small excursion to non-interacting periodic systems with periodicity length L. The previously-used rigid-wall boundary condition will now be replaced by a set of spatial periodicity conditions for the wave function of the system:

ψ(~r ) = ψ(~r + L~ex) = ψ(~r + L~ey) = ψ(~r + L~ez) . (205) The spatial 1-particle energy eigenfunctions are now periodic plane waves (see App. A.1): 1 1 ~ ψ~ (~r ) = exp i[xkx + yky + zkz] = exp(ik ~r ) , k √L3 √L3 · ~2 ~2~k 2 ~2k2 E = (k2 + k2 + k2) = , (206) k 2m x y z 2m ≡ 2m in terms of the quantized wave vectors 2π ~k = ~ν, with ν = 0, 1, 2, . (207) L x,y,z ± ± ··· The remaining steps closely follow the previous discussion. Each unit cube in ~k-space now contains precisely (2s +1)(L/2π)3 = (2s +1) V/(8π3) fully specified quantum states (including spin). In the continuum limit the number of states N(k) with wave vector smaller than k is related to the corresponding volume in ~k-space: k V 4 3 2s +1 3 N(k) = dk′ D(k′) (2s + 1) πk = Vk . (208) ≈ 8π3 3 6π2 Z0 This expression is identical to the result (187) for a particle inside a cube with edges L and rigid walls. As such, also the 1-particle density of states is identical for both types of systems. The most prominent difference between both types of systems is the absence of finite-size effects in the periodic case. This difference vanishes in the limit L . In → ∞ that case we are effectively dealing with a system without explicit periodicity or enclosure. In such scenarios we usually resort to using a periodic system with periodicity length L and subsequently take the continuum limit L (see App. A.2). →∞

Ground state: in the ground state a Fermi gas with periodic boundary conditions occu- pies all 1-particle ~k-states inside a Fermi sphere with radius 6π2 1/3 k = ρ1/3 (209) F 2s +1 N for which N(kF )= N . On the Fermi surface of the Fermi sphere the energy and momen-

tum are given by the Fermi energy EF and the Fermi momentum pF :

~2k2 E = F and p = ~k . (210) F 2m F F

76 2.5.3 Fermi-gas model for conduction electrons in a metal

As a first example of a Fermi gas we consider conduction electrons in a piece of metal. In a metal the electrons of the metal atoms can be divided into two categories. The bulk of all the electrons occupy very narrow (nearly atomic) energy bands, being effectively bound to the atomic nuclei to form metal ions. On the other hand there are the conduction electrons, which occupy the partially filled highest energy band (conduction band). In an individual (separate) metal atom these electrons would be the valence electrons situated in the outer- most atomic shell. In first approximation these conduction electrons can roam freely inside the metal lattice (nearly-free electron model). This phenomenon occurs when the average distance between the metal ions is smaller than the average extension of the atomic orbitals for these conduction electrons. In this oversimplified, semi-classical picture the conduction electrons are kind of squeezed out of their atomic confinement by the metal lattice, as sketched in the picture displayed below where the hatched area represents the conduction 28 29 3 band. For characteristic metal densities of ρNat = 10 –10 metal atoms/m the effective 1/3 volume per metal atom corresponds to the volume of a sphere of radius rs = 3 4πρNat , 10 which is no more than 2– 6 times the Bohr radius a =0.529 10− m. 0 ×

separate atom “edge” macroscopic chain “edge”

...

V (x) V (x)

energy levels energy bands (closely spaced energy levels)

In ﬁrst approximation the conduction electrons form a Fermi gas (electron gas), with the edge of the metal delimiting the macroscopic box (Sommerfeld, 1928).6 In this very crude approximation the explicit electron–electron as well as electron–ion interactions are represented by an averaged constant (attractive) potential V = V < 0. − 0

6As a whole, the combination of metal ions and electron gas is neutral. Such a combination of an electron gas and a complementary collection of positively charged particles is also referred to as a plasma.

77 V

0 L x

EF conduction band

V − 0 metal

As we have seen in 2.5.1 this adds a constant additive contribution NV to the total § − 0 energy of the electron gas. This constant term is not essential for the thermal response of the system. However, it is instrumental in binding the conduction electrons to the metal. Inside the metal lattice the conduction electrons can be considered approximately free, with scattering being suppressed due to the fact that scattering into occupied 1-particle states is forbidden (Pauli-blocked) by the exclusion principle. This eliminates the large majority of all electron–electron and electron–ion collisions. Pauli-allowed collisions, with a suﬃciently large energy transfer, actually provides the mechanism of weak energy contact that allows the gas of conduction electrons to achieve thermodynamical equilibrium inside the metal. This will play a crucial role in understanding the thermal properties of metals, as we will see shortly.

The heat capacity of metals at room temperature.

Consider the heat capacity ∂Etot/∂T V,N of an ideal gas consisting of a constant number N of conduction electrons inside a ﬁxed volume V . In classical statistical mechanics, equipartition of energy tells us that each free particle of an ideal gas contributes 3kB/2 to the heat capacity. For N conduction electrons this would amount to a total classical heat capacity of at least 3NkB/2. This is nowhere near the experimentally measured heat capacities of metals, which turn out to have much smaller values.

Question: can we understand this based on what we have learned so far?

As mentioned above, in ﬁrst approximation the conduction electrons form a Fermi gas with the edge of the metal delimiting the macroscopic box. The particle density of the conduction-electron gas equals the density of metal atoms multiplied by the number of conduction electrons Zv per metal atom. For typical conductors such as copper (Cu) or silver

(Ag) we have Zv = 1 and both densities coincide. A conduction-electron gas at T = 0 has an average kinetic energy per conduction electron of E¯kin =3EF /5, which translates into a

78 classical gas at temperature Tcl =2EF /(5kB)=2TF /5. For instance, silver has a particle density of ρ =5.8 1028 particles/m3 for both metal ions and conduction electrons. By N × means of the table below equation (203) we find the Fermi energy to be E 5.5 eV, F ≈ which corresponds to an effective classical temperature of T 2.5 104 K. By increasing cl ≈ × the thermal energy of the system, conduction electrons can start to occupy excited states with kinetic 1-particle energy E > E . At room temperature, i.e. T 300K, the kin F k ≈ available thermal energy per conduction electron amounts to 3k T /2 0.039 eV E . B k ≈ ≪ F As a consequence, only a small fraction of the conduction electrons is liable to be excited, i.e. conduction electrons with kinetic energy E = E (k T ). In the figure displayed kin F − O B k below this is illustrated by plotting the degree of occupation n(Ekin) of the fully specified

quantum states with given kinetic energy Ekin .

n(Ekin)

1 T =0

0.5 0 < T T ≪ F Ekin 0 EF

The distributions for T =0 and 0 < T T only diﬀer marginally. Later on in this ≪ F chapter it will be proven that at room temperature only a fraction of order Tk/TF of the conduction electrons will participate in the heat conductivity of a metal. This heat conductivity is actually largely determined by the thermal motion of the ions in the metal lattice (phonons) and not so much by the freely moving collective of conduction electrons! This is the reason why you will not be burning your ﬁngers on a piece of metal at room temperature, in spite of the fact that the metal possesses certain characteristic properties of a classical gas at temperature T = (104 K). cl O Note: the above-given Fermi-gas model does not apply to all properties of a metal. For instance, in order to adequately describe the electrical conductivity of metals, the interactions between the conduction electrons and the lattice of metal ions are indispensable. Among other things, these interactions give rise to the band structure in metals and the Cooper-pair binding in low-temperature superconductors.

79 2.5.4 Fermi-gas model for heavy nuclei

Up to now it has only been possible to de- Hahn – Ravenhall – Hofstadter (1956) scribe the collective nuclear forces inside atomic nuclei by means of simplified models. These models parametrize our present experimental and theoretical knowledge. Ca In On the one hand, the radius of the charge V Sb distribution inside the nucleus, i.e. the pro- Co ton distribution, can be inferred from scat- Au Bi tering experiments and spectroscopy. For example: the scattering of electrons off nuclei and the Röntgen spectra of muonic atoms. In the latter case an atomic elec- r [fm] tron is replaced by a muon, which is 200 nucleon distribution in arbitrary units times heavier and therefore has 200 times smaller Bohr orbits. On the other hand, information about the radius of the complete nucleon distribution, involving both protons and neutrons, can be inferred from the scattering of α-particles (helium nuclei) off nuclei (see figure). In that case a relatively sharp energy threshold is observed, marking the minimum energy that the α-particles need in order to overcome the nuclear Coulomb-repulsion and become sensitive to the nuclear forces. By means of these experiments we have learned that the nucleon density is almost constant inside a heavy nucleus:

ρ 0.17 1045 nucleons/m3 mass density 2.8 1017 kg/m3 . (211) N ≈ × ⇒ ≈ ×

This density only has a mild dependence ρ/ρN on the type of nucleus and falls off rel- surface thickness atively quickly in the vicinity of the nuclear surface. Therefore it makes sense to assign to each nucleus a spherical volume with effective radius R, which is defined in such a way that it contains all nucleons nuclear radius when assuming a constant nucleon density ρN . This nuclear radius roughly satisfies the relation r [fm]

1/3 1/3 3 15 R r0 A , with r0 1.12 10− m =1.12fm . (212) ≈ ≡ 4πρN ≈ × The quantity A is the mass number of the nucleus, which is the sum of the number of protons Z (charge number) and the number of neutrons A Z . − 80 Thomas – Fermi model: these observations can be combined into a simple Fermi-gas model (Thomas–Fermi model) for heavy atomic nuclei. In this model the nucleus is comprised of two independent Fermi gases of protons and neutrons that are bound to a sphere of radius R by a constant attractive potential V = V < 0 inside the sphere. This po- − 0 tential well replaces the average short-distance interactions with neighbouring nucleons. The distribution of the two types of nucleons inside the nucleus are in first approximation taken to be uniform, i.e. finite-size effects are neglected. At first sight it may seem non- sensical to “leave out” the actual strong interactions among the nucleons. However, a few properties of nuclei can be explained by Pauli’s exclusion principle alone. By studying the Fermi-gas model we want to figure out which properties fall into this category and which properties require detailed knowledge about the dynamics of the nuclear forces. If we wish, the model can be refined by additionally taking into account the Coulomb repulsion between protons as well as an approximation for the nuclear forces between two nucleons. This obviously makes the model less simple, since the interactions are very sensitive to the actual distance between the two nucleons that participate in the interaction and therefore to their 2-particle spin state. After all, spatial symmetrization/antisymmetrization decreases/increases the average distance between the particles.

V V including Coulomb repulsion

0 R 0 R′ r r

(p) (n) EF EF ∆VCoul/Z V0 − per proton neutrons protons

In the ground state both Fermi gases ﬁll all energy levels up to the Fermi energy, causing the ground-state energy of a heavy nucleus to simply amount to 3 ZE(p) +(A Z)E(n) AV 5 F − F − 0 in this very simple model. So, if we do not account for Coulomb repulsion between the protons the total ground-state energy becomes minimal for Z = A/2 (see exercise 16). The pure Fermi-gas model therefore predicts ground-state nuclei that consist of equal amounts of protons and neutrons:

(p) (n) ρN (203),(211) 3 rel ρ = ρ ======8.25 10− ρ . (213) N N 2 × N

81 The empirical fact that the ratio Z/(A Z) decreases for increasing A-values can only − be understood if we do account for the Coulomb repulsion between the protons (see exercise 16), making it energetically favourable to have slightly more neutrons than protons in

the nucleus. Since the proton mass (Mp) and neutron mass (Mn) are almost identical, both Fermi gases will have roughly the same Fermi energy in the absence of Coulomb repulsion:

~2 2/3 (203),(213) (198) 2 ρN EF ==== 3π EF 38 MeV , (214) 2Mp 2 −−−−−−→ ≈ (203) with corresponding Fermi temperature T = E /k ==== (1011 K). The average kinetic F F B O energy per nucleon then reads 3 E¯ = E 23MeV (215) kin 5 F ≈ and the average binding energy per nucleon E¯ = V E¯ . (216) b 0 − kin By scattering low-energy neutrons oﬀ complex nuclei, the depth of the potential well V0 is measured to be about 40 MeV. The Fermi-gas model therefore predicts an average binding energy per nucleon of 40 MeV 23 MeV = 17 MeV. −

E¯b asymmetric interactions [MeV]

very stable nuclei

surface eﬀects fusion ﬁssion

The experimental data on the binding energy per nucleon (see ﬁgure) follows directly from the diﬀerence between the observed mass of the heavy nucleus and the sum of the rest masses of the individual nucleons: AE¯ = ZM + (A Z)M M c2 , b p − n − 2 2 Mp c = 938.27 MeV and Mn c = 939.57 MeV . (217)

82 The corresponding experimental results for the binding energy per nucleon can be cast into a rather insightful form, known as the semi-empirical mass formula (Bethe –Weizs¨acker formula, 1935):

2 2 ¯ 1/3 Z 2Z A E = aV aS A− aC a − + pair interactions , b − − A4/3 − a A

aV = 15.8 MeV , aS = 18.3 MeV , aC = 0.71 MeV and aa = 23.2 MeV . (218)

The so-called volume coefficient aV is almost completely explained by the exclusion principle. Furthermore, it will be shown in the exercise class that the Fermi-gas model will be less successful in predicting the asymmetry coefficient aa , which quantifies the influence of Z = A/2 on the binding energy. That should not come as a big surprise as we expect 6 nucleon-sensitive interactions such as Coulomb and isopsin interactions to play a role in that quantity. By contrast, the binding-energy effects caused by a uniform charge distribution of protons inside the nucleus are found to agree extremely well with the observed

Coulomb coeﬃcient aC .

Additional remarks: the fact that the total binding energy of a heavy nucleus is approximately proportional to the mass number A is called saturation. In fact the first signs of saturation already set in for relatively light nuclei with A 12. The physical picture ≥ behind this is that the nucleons of heavy nuclei only interact with a small, fixed number of nearest neighbours in view of the short-range nature of the nuclear forces, which fall off exponentially on characteristic distance scales of roughly 1.4 fm. For a closest packing of nucleons the number of nearest neighbours indeed equals 12. This is in sharp contrast with long-range interactions, such as Coulomb interactions, which involve the interactions between all pairs of particles. In that case we expect the binding energy to be proportional to the number of pairs 1 A(A 1), as can be read off from the Coulomb-repulsion 2 − term Z2 in the semi-empirical mass formula. This nuclear saturation effect legitimizes ∝ the point of view that each nucleon in the inner parts of a large nucleus experiences an effective (averaged) nuclear binding potential that does not depend on the total number of nucleons. The saturation effect also predicts the nuclear surface to have a dual influence 1/3 on the binding energy (see the surface term A− = r /R in the semi-empirical mass ∝ 0 formula). First of all, there are finite-size effects that affect the density of states of the Fermi gas, as worked out in exercise 15(ii). Secondly, the particle density has a sharp drop-off in the vicinity of the nuclear surface, causing the nucleons near the edge of the nucleus to have fewer nearest neighbours and consequently less binding energy. The latter aspect gives rise to surface tension, resembling the kind of surface tension that is observed for liquid droplets. For this reason the Fermi-gas model including finite-size effects is also referred to as the liquid-drop model.

83 2.5.5 Stars in the ﬁnal stage of stellar evolution

A star can be regarded as a many-particle system for which the gravitational pressure is in equilibrium with the thermal and radiation pressure. During the stellar evolution various nuclear-fusion stages can be identified, with the fusion fuel being consumed step- by-step (and shell-by-shell) from the inside out. In the final stage of stellar evolution the fusion fuel gets exhausted in the stellar core, where lowly reactive nuclei such as 12C- and 16O-nuclei have amassed. If the fusion reactions effectively cease in the stellar core, then so does the radiation pressure. With one component of the compensatory pressure gone, the star will start to implode in an attempt to find a new equilibrium. While the star implodes the outer layers of stellar matter are ejected, leaving behind a stellar core with a thin atmosphere. Depending on its mass, this stellar core can go through different phases.

Phase 1: Fermi-gas model for white dwarfs. During the implosion the thermal (kinetic) pressure increases in the stellar core, where the matter is being compressed. Assume, for convenience, that the stellar core is comprised of one characteristic type of atom with charge number Z and mass number A = 2Z . If the stellar matter is compressed to the point that the distance between the atomic nuclei

becomes smaller than the Bohr radius aZ = a0/Z of a 1-electron atom consisting of a nucleus with charge Ze and one electron, then the atoms will be compressed into full ionization (pressure ionization). In ﬁrst approximation the electrons that are “squeezed out of the atoms” will form an electron gas inside a spherical volume.7 For atoms with A = 12 –16 this phenomenon occurs when the mass density inside the stellar core exceeds the 107 kg/m3 barrier. A star that reaches a new stable equilibrium during this electron- gas phase is called a white dwarf.

Let’s assume the electron density of the electron gas to be uniform, which can be replaced by a more realistic density proﬁle in a reﬁned version of the model (see exercise 18). This constant density can be translated by means of the plasma relation

(e) (nuclei) ρM ρN = ZρN = Z (219) AMp

in terms of the uniform mass density ρM of the stellar core. In this expression AMp is the mass per characteristic atomic nucleus and Z indicates how many electrons are released 7 11 3 per atom. For typical white-dwarf mass densities ρM = 10 –10 kg/m , the electron den- (e) 33 37 3 sity is given by ρN = 10 –10 electrons/m and the corresponding Fermi temperature by 8 10 TF = 10 –10 K (remembering that A =2Z ). The typical thermodynamic temperature T< 107 K of a white dwarf is (much) smaller than the Fermi temperature of the electron gas, which therefore eﬀectively will be in the ground state.

7Just as in 2.5.3 the electrons and ions (nuclei) together form a plasma. §

84 Since the kinetic energy of the electrons substantially r+dr exceeds the thermal energy of the much heavier nuclei, we have to look for a stable equilibrium condition based exclusively on the kinetic energy of the r electron gas at T = 0 and the gravitational potential of the star. The gravitational potential is obtained straightforwardly by building up the star in spherical layers

R R 4 G 16 E = dr 4πr2ρ πr3ρ N = π2ρ2 G drr4 grav − M 3 M r − 3 M N Z Z 0 0 4 2 3G 3G M 2 = πR3ρ N = N . (220) − 3 M 5R − 5R

Here M and R are the mass and radius of the imploded star, whereas GN is Newton’s gravitational constant. The total energy of the star is now approximately given by

2 ~2 2 (199) 3 3GN M (198) 3 2 (e) 2/3 3GN M ET ======NEF ==== N (3π ρN ) 5 − 5R 10me − 5R

2 2 2 (219) M 3~ 3π Zρ 2/3 3G M ======Z M N AMp 10me AMp − 5R 2/3 ~2 5/3 5/3 2 5/3 2 ρM =M/V 3 9π Z M 3GN M M 3GN M ======2 a 2 . 10 4 me AMp R − 5R ≡ R − 5R

The corresponding equilibrium condition reads dET /dR = 0, which yields

5/3 2 3 2aM 3GN M 3 10a A=2Z 50 3 3 + 2 = 0 MR = ==== 7.30 10 kg m . (221) − R 5R ⇒ 3GN × In this way we have established a relation between the mass and radius ( 104 km) of ∼ a white dwarf. This relation runs counterintuitive to our gut feeling that a larger star should necessarily be a heavier star. However, this counterintuitive relation is conﬁrmed by astronomical observations. In fact we are dealing here with a pure quantum mechanical phenomenon caused by degenerate quantum matter that manifests itself in a directly measurable way in an object of astronomical dimensions!

Question: Is the non-relativistic approach actually viable here?

For a mass density of ρ = 107 kg/m3 we have ρ(e) = 3 1033 electrons/m3 ρrel , as M N × ≪ N can be extracted from the table below equation (203). At “low” mass densities the non- relativistic approach is indeed perfectly acceptable. However, at “high” mass densities 11 3 relativistic corrections are unavoidable. For a mass density of ρM = 10 kg/m the other

85 extreme is reached, since ρ(e) = 3 1037 electrons/m3 ρrel . In that case the Fermi gas N × ≫ N should be treated (ultra) relativistically. The ultra-relativistic treatment, which implies neglecting the electron mass with respect to the kinetic energy, gives rise to a completely diﬀerent type of equilibrium condition (see exercise 18): stable white dwarfs have an upper limit MC on the mass. This upper limit of roughly 1.4 solar masses is called the Chandrasekhar limit.

radius versus mass for celestial bodies

Phase 2: Fermi-gas model for neutron stars. Heavy stars can pass phase 1 without reaching a stable equilibrium. Such a star will continue to implode. For increasing mass density electrons can be “captured” by the protons through the inverse β-decay process

e− + nucleus(A, Z) ν + nucleus(A, Z 1) , (222) → e − with the approximately massless neutrinos escaping (evaporating) and thereby cooling the star. The physical mechanism behind this inverse β-decay process looks as follows. On the right-hand side of the equation the nuclei have less total binding energy in view of the disturbed balance between the number of protons and neutrons. For suﬃciently high mass densities this energy increase will be compensated by an energy reduction caused by lowering the number of electrons in the electron gas. At a mass density of roughly

86 15 3 ρM = 10 kg/m the kinetic pressure of the neutrons will become more important than the pressure of the electron gas. At even higher densities in the range of 1017 –1018 kg/m3 this will eventually result in a Fermi gas that is predominantly comprised of neutrons (neutron gas). A star that reaches stability during this neutron-gas phase is therefore called a neutron star. The main diﬀerences with the Fermi gas of phase 1 are summarized by two simple substitutions: Z A and m M M . Apart from that every- → e → n ≈ p thing else stays the same. As a result, the a-parameter in equation (221) will receive 5/3 3 an extra factor (m /M )(A/Z) = (10− ). This intrinsic factor of 1000 reduction for e p O the characteristic radii of neutron stars is indeed conﬁrmed by astronomical observations (R 10 km). Again the ultra-relativistic treatment gives rise to an upper limit on the ∼ mass of a neutron star. This upper limit has to be taken with a pinch of salt, though. For mass densities of the order of the nuclear mass density 2.8 1017 kg/m3 we also expect × the dynamics of the strong nuclear force to play an important role.

2.6 Systems consisting of non-interacting particles

As a second application of ensembles in thermal equilibrium, we try to derive the statistical properties of systems consisting of a large number of non-interacting particles. We assume the corresponding 1-particle energy spectrum to be discrete, as is for instance the case for systems that are contained inside an enclosure or trapped in a potential well. Three fundamentally diﬀerent scenarios can be identiﬁed.

(A) The particles are distinguishable, for example by means of their positions. This will give rise to the so-called Maxwell–Boltzmann statistics.

(B) The particles are indistinguishable identical bosons. This will give rise to the so-called Bose–Einstein statistics.

(C) The particles are indistinguishable identical fermions. This will give rise to the so- called Fermi–Dirac statistics.

For deriving the statistical distributions in scenarios (B) and (C) we will use the grand- canonical ensemble approach of 2.4.4. The Maxwell–Boltzmann distribution can then be § obtained by taking the classical limit. Since the particles are assumed to be non-interacting, we choose to work with an occupation-number representation based on a complete set of 1-particle observables that includes the 1-particle Hamilton operator. The total Hamilton operator and total number operator of the many-particle system have a simple additive form in this representation:

Nˆ = nˆj and Hˆtot(Nˆ) = Ej nˆj , (223) j j X X with E E E the ordered fully speciﬁed 1-particle energy spectrum. The 1 ≤ 2 ≤ 3 ≤···

87 statistical properties of the ensemble can be speciﬁed by the average occupation numbers

n¯k = [ˆnk] , since

N¯ = n¯j and E¯tot = Ej n¯j . (224) j j X X For the grand-canonical density operatorρ ˆ = exp βHˆ (Nˆ) αNˆ / (α, T ) this implies − tot − Z ρˆaˆ† = exp( βE α)ˆa† ρˆ . (225) k − k − k

Proof: consider the operator Oˆ(λ) exp(λAˆ)â† exp( λAˆ), with the observable in the ≡ k − ˆ ˆ ˆ ˆ (223) exponent given by A βHtot(N) αN ==== j (βEj + α)ˆnj . With the help of (16) ≡ − − − A,ˆ aˆ† ==== (βE + α)â† we can derive for arbitrary real λ that k − k k P d Oˆ(λ) = exp(λAˆ) A,ˆ aˆ† exp( λAˆ) = (βE + α)Oˆ(λ) dλ k − − k Oˆ(λ) = exp λ[βE + α] Oˆ(0) : exp(λAˆ)â† exp( λAˆ) = exp λ[βE + α] aˆ† . ⇒ − k k − − k k The operator identity in equation (225) is then obtained by simply inserting λ = 1 and subsequently right-multiplying the result byρ ˆ = exp(Aˆ)/ (α, T ). Z The desired statistical distribution for identical particles then becomes

(17),(137) (225) n¯ ==== Tr(ˆρaˆ† aˆ ) ==== exp( βE α) Tr(ˆa† ρˆaˆ ) = exp( βE α)Tr(ˆρaˆ aˆ† ) k k k − k − k k − k − k k

(26),(27) (139) ==== exp( βE α) Tr ρˆ[1ˆ nˆ ] ==== (1 n¯ ) exp( βE α) − k − ± k ± k − k − 1 n¯ = , (226) ⇒ k exp(βE + α) 1 k ∓ where the minus (plus) sign refers to bosons (fermions). In the limit α the identity of →∞ the particles does not matter anymore and the quantum mechanical distributions smoothly approach the well-known classical Maxwell–Boltzmann distributionn ¯ = exp( βE α). k − k − In this asymptotic limit 1-particle quantum eﬀects dominate the statistics as a result of the suppression factor exp( αN) in the grand canonical density matrix. − All this can be summarized by the following distribution for the average occupation number belonging to a fully speciﬁed 1-particle energy level with energy E :

1 n¯(E) = , with γMB = 0 , γBE = 1 and γFD = 1 . (227) exp(βE + α)+ γ −

Note: the temperature and fugacity parameters β and α are not independent of each

other in view of the conditions (181) for the ensemble averages E¯tot and N¯ .

88 The quantum mechanical distributions 8 n¯(E) 7

6 Bose–Einstein

5 Maxwell–Boltzmann

Fermi–Dirac classical 1 domain

0 -2 -1 0 1 2 3 quantum mechanical domain βE +α

The characteristic properties of these distributions can be read oﬀ directly from the plots displayed above.

The three distributions approach each other for large values of βE + α. For instance • this applies to the high-energy limit or to situations for which α is large. In such situations the identity of the particles does not matter anymore and quantum mechanical statistics eﬀectively merges with classical Maxwell–Boltzmann statistics.

As anticipatedn ¯(E) [0, 1] for the Fermi–Dirac distribution. • ∈ For the Bose–Einstein distribution we have to demand that βE + α > 0 for each • energy E, in order to guarantee thatn ¯(E) 0. Hence, α is bounded from below ≥ by α> βE , with E the 1-particle ground-state energy. − ground ground

The Fermi energy: in the low-temperature limit T 0 ( β ) the Fermi–Dirac → → ∞ distribution approaches the distribution

β 1 if E < EF n¯(E) →∞ , −−→   0 if E > EF

for which all energy states are occupied up to the maximum energy EF = lim α/β . − β For a Fermi gas this maximum energy of course coincides with the definition for the→∞ Fermi energy that was introduced in 2.5. § 89 Freeze out of bosonic degrees of freedom: the fact that βE + α βE + α> 0 ≥ ground for the Bose–Einstein distribution has as direct consequence that n¯(E) exp(βE + α) 1 = ground − exp β [E E ] . n¯(E ) exp(βE + α) 1 ≤ − − ground ground − If an energy gap ∆E is present between the 1-particle ground state and the first excited state, then thermal excitations are heavily suppressed for temperatures k T ∆E . Such B ≪ a scenario for instance applies to a harmonic oscillator with characteristic frequency ω for temperatures k T ~ω (see exercise 14). In that case we speak of an effective freeze B ≪ out of the corresponding bosonic degree of freedom since the excited states containing one or more bosonic energy quanta ~ω will be scarcely populated. The effective freeze out of such a degree of freedom goes hand in hand with a reduced thermal response (e.g. heat conduction) of the system. Applied to complex systems such as molecular gases or crystal lattices, this phenomenon can occur step-by-step: for decreasing temperature more and more vibrational modes of the complex system will freeze out.

Bose – Einstein condensation: if βE + α 0 in the Bose–Einstein distribution, then ↓ the corresponding average occupation number diverges. This sharp rise in the degree of occupation is unique for the bosonic distribution and is responsible for the physical phenomenon of Bose–Einstein condensation. The fact that bosons have an increased probability to occupy the same quantum state can result in a noticeable (macroscopic) fraction of the particles occupying the ground state already at ﬁnite temperatures. Whether this phenomenon actually occurs or not depends on the properties of the system considered.

2.7 Ideal gases in three dimensions

An important example of non-interacting many-particle systems in thermal equilibrium is provided by the ideal gases. These systems consist of non-interacting particles of mass m and spin s that are contained inside a macroscopically large enclosure with ﬁxed volume V . We assume the enclosure to contain a very large, approximately constant number N of free identical particles. The 1-particle energy spectrum pertaining to such an enclosed system is derived in 2.5. In view of the macroscopic volume and the very large number § of particles, we can in most situations8 resort to taking the continuum limit (189) for the

1-particle density of states D(Ekin):

2s +1 2m 3/2 V E1/2 if E 0 2 ~2 kin kin D(E ) = 4π ≥ . (228) kin    0 if Ekin < 0

8This actually does not hold for ideal gases consisting of identical bosons at very low temperatures (see the discussion of Bose–Einstein condensation in 2.7.2). § 90 The average number of gas particles with kinetic energy on the interval [Ekin , Ekin + dEkin] is then given by n(Ekin, T ) dEkin , with

(227) D(Ekin) n(Ekin, T )=n ¯(Ekin)D(Ekin) ==== . (229) exp(βEkin + α)+ γ

The quantity α is determined by the constant particle density ρN = N/V :

∞ 3/2 ∞ 1/2 1 (228),(229) 2s +1 2m Ekin ρN = dEkin n(Ekin, T ) ======2 2 dEkin V 4π ~ exp(βEkin + α)+ γ Z Z0 −∞

∞ x=βE ,(166) 2s +1 2mk T 3/2 x1/2 ======kin B dx , (230) 4π2 ~2 exp(x + α)+ γ Z0 where the integral is a strictly decreasing function of α. This last observation will play an important role in the phenomenon of Bose–Einstein condensation discussed in 2.7.2. § By means of the thermal de Broglie wavelength

h λT = characteristic thermal quantum mechanical length scale (231) ≡ √2πmkBT

equation (230) can be rewritten as

2 ∞ x1/2 ρ λ3 /(2s+1) = dx , (232) N T √π exp(x + α)+ γ Z0

which links α directly to the temperature. The size of the right-hand side of this equation will determine whether the system is in the quantum mechanical or classical regime.

Quantum mechanically speaking we are dealing with a classical gas if the average 1/3 distance between the particles satisﬁes ρN− λT , which implies that the ≫ 3 number of particles in the characteristic thermal volume λT is very small and that the individual wave packets of the particles therefore eﬀectively do not 1/3 overlap. For ρ− = (λ ) the gas is in the quantum mechanical regime. N O T

In the same way we can obtain the average total energy density U = E¯tot/V of the ideal gas:

∞ 3/2 ∞ 3/2 1 (228),(229) 2s +1 2m Ekin U = dEkin Ekin n(Ekin, T ) ======2 2 dEkin V 4π ~ exp(βEkin + α)+ γ Z Z0 −∞

∞ x=βE ,(166) 2s +1 2mk T 3/2 x3/2 ======kin B k T dx , 4π2 ~2 B exp(x + α)+ γ Z0 91 which can be rewritten by means of equation (230) as

∞ x3/2 ∞ x1/2 U = ρN kBT dx dx . (233) exp(x + α)+ γ , exp(x + α)+ γ Z0 Z0

From this we can derive the heat capacity per unit of volume

∂U CV = . (234) ∂T V,N 2.7.1 Classical gases

If α 1, then exp(x + α) γ and all three types of distributions coincide. In that case ≫ ≫ we speak of classical gases. The corresponding condition (232) for α simpliﬁes to

2 ∞ ρ λ3 /(2s+1) dx x1/2 exp( x α) = exp( α) (235) N T ≈ √π − − − Z0

∞ 1/2 1 if we employ the integral identity dx x exp( x) = Γ(3/2) = 2 √π. For α 1 the 0 − ≫ right-hand side of this equation indeedR becomes small, which duly puts the system in the classical regime. This scenario occurs when the particle density is sufficiently low and/or the temperature sufficiently high. For instance, this applies to gases at room temperature and room pressure, for which quantum mechanical effects due to the identity of the particles indeed do not matter.9 As anticipated from classical kinetic gas theory, classical gases can be described perfectly well by Maxwell–Boltzmann statistics. The corresponding average total energy density of equation (233) then simply reads 3 U ρ k T, (236) ≈ 2 N B

∞ 3/2 3 if we employ the integral identity dx x exp( x) = Γ(5/2) = 4 √π . From this follows 0 − the typical heat capacity per unit ofR volume of a classical gas:

∂U 3 NkB CV = = , (237) ∂T V,N 2 V in full agreement with the principle of equipartition of energy.

9The Fermi gases as described in 2.5 do not fall into this category in view of the high particle density. § In the classical era one therefore had no reason to interpret such systems as gases.

92 2.7.2 Bose gases and Bose–Einstein condensation (1925)

For ideal gases that consist of identical bosons we know that there is the additional condition α > βE = βE 0 to be satisﬁed, in order to make sure that − ground − ν=√3 ≈ βE + α > 0 and therefore guarantee that n(E , T ) 0. Equation (232) now implies kin kin ≥ that 2 ∞ x1/2 ρ λ3 /(2s+1) = f (α) , f (α) dx (238) N T 1/2 1/2 ≡ √π exp(x + α) 1 Z0 −

with the Bose gas being in the quan- 4

tum mechanical regime if f1/2(α)= (1). f (α) O 1/2 T

ﬁnite value f1/2(0) = 2.612. Something 2

special happens as soon as the tempera- 3 T >T0 ρN λ ture is lowered to the critical temperature T (2s+1) T0 for which equation (238) has a solution (indicated by the intersection with one of 1 the horizontal dashed curves in the plot) that coincides with the minimum value of

α and therefore with the maximum value 0 of f (α): 0 1 2 3 4 5 1/2 α

(231),f (0)=2.612 ~2 2/3 3 1/2 2π ρN ρN λT0 /(2s+1) = f1/2(0) ======T0 = . (239) ⇒ mkB 2.612 (2s + 1)

For T < T0 suddenly no solution to equation (238) can be obtained anymore, indicating that apparently something has gone wrong!

In order to properly describe the low-temperature behaviour of Bose gases, a very careful treatment of the low-energy 1-particle states is paramount. After all, compared with distinguishable particles the indistinguishable identical bosons have an increased tendency to occupy the 1-particle ground state. From the continuum limit (189) we can read oﬀ that D(0) = 0, whereas in fact 2s+1 diﬀerent 1-particle states exist that have the lowest 1-particle energy E = E 0. The minimum requirement for correcting the ground ν=√3 ≈

93 1-particle density of states D(Ekin) thus reads

D(E ) D¯(E ) = D(E ) + (2s+1)δ(E ) , (240) kin → kin kin kin so that

N0 3 (227) 2s +1 ρ = + λ− (2s+1)f (α) , N = (2s+1)¯n(0) ==== . (241) N V T 1/2 0 exp(α) 1 −

In this expression N0 represents the average number of bosons in the 1-particle ground state. This ground-state occupation number can be neglected with respect to N, unless α = (1/N) 0. As such, the density of particles in the 1-particle ground state starts O ≈ to play a prominent role if the temperature is lowered to the critical temperature T0 for which α 0. By taking the ratio with respect to ρ , equation (241) can be rewritten in ≈ N terms of the critical temperature:

3 3/2 N0 (2s+1)f1/2(α) f1/2(α) λT0 f1/2(α) T = 1 3 = 1 3 = 1 . (242) N − ρN λT − f1/2(0) λT − f1/2(0) T0 The last term on the right-hand side represents the fraction of bosons that occupy excited 1-particle states.

T < T0 : below the critical temperature a noticeable fraction of all bosons should be in the ground state, since α cannot become smaller than 0 and therefore at most a fraction 3/2 (T/T0) < 1 of all bosons can occupy excited 1-particle states. In that case we say that the ground state is occupied macroscopically. According to equation (241), getting a fraction N /N = (1) of all bosons to occupy the ground state requires α = (1/N) 0. 0 O O ≈ To very good approximation the fraction of bosons in the ground state is therefore given by N T 3/2 0 1 . (243) N ≈ − T0 This observable tendency of the identical bosons to occupy the same quantum state is called Bose–Einstein condensation (BEC). This condensation takes place in momentum space and should not be confused with the more traditional meaning of the word condensation as a transition from a gaseous phase to a liquid phase. To increase the amount of condensation requires lower temperatures or higher densities.10 The latter statement follows from the fact that according to equation (239) T ρ2/3. The condensate of bosons in the ground 0 ∝ N state forms a degenerate Bose gas, which will not exert pressure nor contribute to the heat capacity. At low temperatures only the fraction of bosons outside of the condensate will

10Note: it is far from easy to realize Bose–Einstein condensation in the lab. At high densities any interaction among the bosons tends to completely swamp the symmetrization eﬀects. The density should therefore be kept low, causing T0 to approach absolute zero (microkelvins or less).

94 2

CV ρ k N B classical 1.5 limit (T/T )3/2 ∝ 0 1

0.5

0 0 0.5 1 1.5 2 2.5 3

T/T0

contribute to the pressure and heat capacity of the Bose gas, giving rise to an additional 3/2 factor (T/T0) in the temperature dependence of these quantities (as the plot for the heat capacity reveals). More details about this are given in the lecture course on Statistical Mechanics. In 2.7.4 we will brieﬂy discuss why it took another 70 years to actually realize § the ﬁrst gaseous Bose–Einstein condensates in the lab.

T > T0 : in that case α has to diﬀer noticeably from 0 to avoid that the ratio N0/N in equation (242) becomes negative. Automatically only a negligible fraction of the bosons occupy the ground state.

Occupation of the first excited state for T < T0 : in order to say something sensible about this we consider the 1-particle energy spectrum (184) for a particle in a cubic box. For the energy difference between the ground state with ν = √3 and the first excited state with ν = √6 we can derive that

2/3 ~2 2 3 2 3 π V =L 2/3 T0 3h ρN (231) 2/3 T0 3π 3 2/3 β (E E ) = ==== N − ==== N − (ρ λ ) ex ground 2 N T0 − 2mL kBT T 8mkBT0 T 4

(239) 2/3 T0 3π 2/3 2/3 ==== N − 2.612 (2s+1) = (N − T /T ) . T 4 O 0 The maximum average occupancy of the ﬁrst excited energy level is obtained for the minimum value for α, i.e. for α βE : ↓ − ground 1 1 n¯(E ) = (N 2/3) N. ex ≤ exp β [E E ] 1 exp [N 2/3 T /T ] 1 ≤ O ≪ ex − ground − O − 0 − So, solely the 1-particle ground state can achieve macroscopic occ upancy in a 3-dimensional Bose gas! The adjacent ﬁrst excited energy level can contain many particles, but nevertheless will not be able to reach macroscopic occupancy.

95 The phase transition at T = T0 : as explained on page 92, the quantum mechanical domain of an ideal gas is reached if ρ λ3 = (1). In that case the wave packets of the N T O individual particles will overlap and therefore the quantum mechanical indistinguishability of the identical particles becomes relevant. In the vicinity of the critical temperature T0 this is indeed the case. The remarkable aspect of T0 is that the gas ondergoes a rapid transition (phase transition) at that temperature.

For an arbitrary temperature T < T0 an ideal Bose gas is comprised of a ﬁxed, saturated number of particles in excited states. The distribution of these particles has a canonical form: it is described by a Bose–Einstein distribution with α = 0. The remaining particles occupy the ground state, which thereby achieves macroscopic occupancy. The presence of the condensate is a direct consequence of the underlying particle conservation law.

Proof: for T < T0 we know that α vanishes, which implies that the excited states are maximally occupied according to a canonical distribution. From equation (243) we know 3/2 that the number of particles that do not occupy the ground state is given by N (T/T0) . When increasing the number of particles by a factor f while keeping the temperature constant, we get

(239) N fN === T f 2/3 T and N (T/T )3/2 N (T/T )3/2. → ⇒ 0 → 0 0 → 0 The number of particles outside the condensate indeed remains constant. We could actually have read this oﬀ directly from equation (239), since

3/2 (231),(239) 3 N (T/T0) ======2.612 (2s+1)V/λT is indeed independent of the particle density and merely counts the number of times the 3 characteristic thermal volume λT ﬁts into the system volume V . The fact that we have added additional particles merely shows up in the occupancy of the ground state! One could say that this condensate “does not occupy volume”, as it does not exert pressure, and therefore can be adjusted freely to account for the correct total particle density. On this ground we anticipate that the Bose–Einstein distribution will have a (N) statistical uncertainty in the occupancy of the ground state, O as will be shown in exercise 19. Moreover, the presence of the condensate manifests itself in long-range correlations (long-range order) in view of the corresponding macroscopic de Broglie wavelength. The high-occupancy condensate behaves approximately as a (pseudo classical) matter wave that can be described quantum mechanically by means

of a complex function (order parameter) ψ0 = √N0 exp(iφ0).

96 Quantum mechanical indistinguishability and Bose–Einstein condensation: the reason behind the absence of condensation for distinguishable particles resides in the

number of possible ways to excite particles from the ground state. Assume N0 spin-0

particles to occupy the ground state. Then there are 3N0 diﬀerent ﬁrst excited states if the particles are distinguishable and merely 3 if the particles are indistinguishable.

ground state excited state

0 E (3 ) 1 E (3 ) ex × ex ×

N E N 1 E 0 ground 0 − ground

For large enough N0 it becomes entropically more favourable for distinguishable particles to be excited because of the kB ln(3N0) gain in entropy. For indistinguishable particles the gain in entropy is substantially smaller and independent of N0 , i.e. kB ln(3), which lifts the entropic obstruction for having a large number of particles in the ground state.

Bose–Einstein condensation versus D(Ekin) for low energies: the previous indistinguishability argument is actually not able to guarantee the occurrence of a condensate. If the continuous density of states is not small for low-energy 1-particle states, then there are nevertheless ample of ways to excite a particle with a minimum gain of energy. This can keep the system from actually achieving a macroscopic occupation of the ground state. In the above-given derivation for a 3-dimensional Bose gas it was crucial for the occurrence of Bose–Einstein condensation that the strictly decreasing integral f1/2(α) remained ﬁnite when α reached its minimal value. In exercise 20 it will be investigated how the formation of a condensate depends on the dimensionality of the system and on the energy–momentum relation of the particles. The results of this investigation can be summarized as follows:

Bose–Einstein condensation is indeed not possible if D(Ekin) does not vanish

for Ekin = Eground : in that case ∞ dEkin D(Ekin)/ exp(βEkin + α) 1 is not −∞ − ﬁnite for α βEground and the continuous density of states D(Ekin) does not ↓ − R require being corrected.

The role of the particle conservation law: in the absence of particle conservation there is no reason for a condensate to form. A photon gas in thermal equilibrium with an atomic heat bath is an example of this (see 4.3.2). As a result of being absorbed by § the atoms, the photons actually do not show up as particles in the atomic heat bath and therefore they exclusively contribute to the thermodynamic balance of energy. In that case the entropy of the heat bath actually does not depend on the number of photons.

On the other hand, a canonical ensemble with a ﬁxed number of particles is able to give rise to Bose–Einstein condensation, as is for instance the case for low-energy photons in the

97 curved-mirror microresonator of exercise 21. In that case the ﬁxed number of particles acts as a boundary condition on the occupation numbers of the 1-particle energy levels, which resembles the action of keeping the ensemble average N¯ constant in a grand-canonical ensemble.

2.7.3 Fermi gases at T = 0 (no exam material) 6 For ideal gases that consist of identical fermions there can be no condensation phenomenon, since maximally 2s + 1 fermions can occupy the ground state as a consequence of the exclusion principle. Therefore there is no need to correct the continuous density of states

D(Ekin). We simply obtain

2 ∞ x1/2 ρ λ3 /(2s+1) = g (α) , g (α) dx . (244) N T 1/2 1/2 ≡ √π exp(x + α)+1 Z0

The low-temperature domain (degenerate Fermi gases): for Fermi gases we speak 3 of low temperatures if ρN λT is large, which implies that α has to be strongly negative in order to satisfy equation (244). In that case the following integral approximation can be used for a function G(x) that grows slower than exponentially for large x:

α ∞ − G(x) 2 dx 1+ [α− ] dxG(x) , (245) exp(x + α)+1 ≈ O Z0 Z0

provided that the integral on the right-hand side exists. This approximation follows from 1 the fact that the function n(x) [exp(x+α)+1]− effectively behaves like a step function ≡ for large α, with a deviation that is symmetric around x = α (zie plotted curve). This − − deviation is only significant for a limited range of x-values: x + α =(E µ)/k T = (1), − B O which corresponds to kinetic energies that only differ from the Fermi energy by (k T ). O B n(x)

step function approximation 1

0.5 actual distribution x 0 α −

x + α = (E µ)/k T = (1) − B O

98 For a degenerate Fermi gas equation (244) leads to the following approximate relation

between α, ρN and T :

3 4 3/2 2 ρ λ /(2s+1) ( α) 1+ [α− ] N T ≈ 3√π − O 2 2 (231) ~ 6π ρ 2/3 (198) E (201) T === α N ==== F ==== F . (246) ⇒ − ≈ 2mkBT 2s +1 kBT T 2 2 2 This relation is valid up to (α− )= (T /T ). In E and T we immediately recognize O O F F F the Fermi energy and Fermi temperature as derived for T = 0 Fermi gases in equations (198) and (201). In accordance with the examples discussed in 2.5 the domain of degen- § erate Fermi gases corresponds to α T /T 1, which can be reached by lowering the − ≈ F ≫ temperature or increasing the density and (related to that) T . Only a (T/T ) fraction F O F of the particles will participate in the thermal response of a degenerate Fermi gas. This can be read oﬀ from equation (233) by employing the integral approximation (245):

2 5/2 5 ( α) 2 (246) 3 2 2 U ρ k T − 1+ [α− ] ==== ρ k T 1+ [T /T ] ≈ N B 2 ( α)3/2 O 5 N B F O F 3 − ∂U CV = = (ρN kBT/TF ) . (247) ⇒ ∂T V,N O More details will be given in the lecture course on Statistical Mechanics. Contrary to Bose gases, in a degenerate Fermi gas almost all particles will contribute to the kinetic energy and pressure of the gas, regardless of the temperature.

Chemical potential: in statistical mechanics α is usually rewritten in terms of the so- called chemical potential

α (166) µ ==== αk T . (248) ≡ − β − B

For a degenerate Fermi gas we have µ E , so that ≈ F

(227) 1 1 n¯(E) ==== . (249) exp β [E µ] +1 ≈ exp β [E E ] +1 − − F For a completely degenerate Fermi gas at T = 0 all states will be occupied up to the Fermi energy:

1 if E < EF n¯(E) = ,   0 if E > EF in accordance with the deﬁnition given in 2.5. §

99 Example: Pauli paramagnetism of a degenerate electron gas. Consider an ideal electron gas with constant volume V and constant number of electrons N. Assume that T T , which implies that the chemical potential µ of the gas satisfies ≪ F βµ = µ/k T 1. Subject this system to a very weak constant homogeneous magnetic B ≫ field in the z-direction, ~ = ~e . In spin space this gives rise to an extra interaction B B z 2µ ˆ spin ~ˆ ~ ~ˆ B ˆ H = = ~e = B S (0 < µ kBT µ) , B − MS · B − B MS · z ~ z B B ≪ ≪ in terms of the Bohr-magneton µB . As explained on page 73, the magnetically induced spin interaction leads to a splitting of the kinetic 1-particle energy levels, thereby lifting the spin degeneracy. The average number of electrons with spin parallel (N¯+) and antiparallel (N¯ ) to the magnetic field are given by −

∞ ∞ 1 D (E ) (196) 2 D(E µB ) N¯ = dE ± ± ==== dE ± ∓ B , ± ± exp β [E µ] +1 ± exp β [E µ] +1 Z ± − µZ ± − −∞ ± B B with E the 1-electron energy for states with spin component ~/2 along the z-axis and ± ± with D(Ekin) the kinetic 1-particle density of states (228) for m = me and s =1/2. By means of the change of variable E = Ekin µ this can be simpliﬁed to ± ± B B

3/2 ∞ 1/2 V 2me Ekin N¯ = dEkin , (250) ± 2 ~2 4π exp β [Ekin µ µB ] +1 Z0 − ± B with Ekin the kinetic 1-electron energy. This allows us to derive the following additive expressions for the (constant) electron density and the magnetization density, i.e. the average magnetic moment per unit of volume in the direction of the magnetic ﬁeld:

N N¯+ + N¯ ρ = = − , N V V

1 2µ (53) µ ~ˆ (1) ~ B ~ˆ (1) B ¯ ¯ MS [ S ] = [S tot ~ez ] ==== (N+ N ) . (251) ≡ V M tot · B − ~V · − V − − B In the limit µ µ this can be simpliﬁed by means of the integral approximation (245): B B≪ 1 2m 3/2 1 2m µ 3/2 ρ e [µ µ ]3/2 + [µ + µ ]3/2 e , N ≈ 6π2 ~2 − B B B B ≈ 3π2 ~2 µ 2m 3/2 1 2m µ 3/2 3µ2 M B e [µ + µ ]3/2 [µ µ ]3/2 e B B . S ≈ 6π2 ~2 B B − − B B ≈ 6π2 ~2 µ In lowest-order approximation we have µ E and ≈ F

2 3ρN EF = kBTF dMS 2 3ρN MS µB ======χS (T ) = µB . (252) ≈ 2EF B ⇒ ≡ d =0 2kBTF B B 100 The paramagnetic susceptibility χ (T ) of the electron gas is constant for T T . This S ≪ F low-temperature phenomenon is called Pauli paramagnetism. It stresses that a degenerate 1 Fermi gas does not obey Curie’s classical law χ (T ) T − for weak magnetic fields. This S ∝ is caused by the effective quantum mechanical interaction induced by the fermionic antisymmetrization procedure. In the classical high-temperature limit the magnetic allignment of the magnetic moments is counteracted by thermal disorder. This leads to a magnetization density that depends on the ratio of the magnetic interaction energy µ and the B B characteristic thermal energy kBT , resulting in a net 1/T temperature dependence of the paramagnetic susceptibility. At low temperatures only a (T/T ) fraction of the electrons O F take part in thermal excitations, effectively replacing the 1/T temperature dependence by a temperature-independent factor 1/TF . The exclusion principle is more effective at coun- teracting magnetic allignment than thermal disorder, since the characteristic temperature

TF of a degenerate Fermi gas can largely exceed the (ambient) temperature T .

2.7.4 Experimental realization of Bose–Einstein condensates

Superfluid helium: 4He-atoms consist of two protons, two neutrons and two electrons, which implies that the corresponding total spin is integer. The 4He-atoms can thus be regarded as bosons. They satisfy Bose–Einstein statistics and can therefore give rise to the formation of a Bose–Einstein condensate at low temperatures. As argued in 1.6.4 –1.6.6, § this condensate plays an essential role in the phenomenon of superfluidity in liquid 4He at temperatures below 2.2 K. The rare 3He-isotope consists of two protons, one neutron and two electrons, which implies that the corresponding total spin is half integer. The 3He-atoms can thus be regarded as fermions. They satisfy Fermi–Dirac statistics and can therefore not give rise to a similar type of condensate as 4He-atoms. Pairs of 3He-atoms, however, can behave bosonically. As a result of pairing effects, also 3He can display superfluidity, but only at temperatures below 2 mK.

Ultracold dilute gases: for long the existence of superfluidity was seen as an indica- tion that it should be possible to realize a gaseous Bose–Einstein condensate in the lab. However, that was all that could be said, since the observed superfluidity involved flu- ids rather than an ideal or weakly-interacting gas. It took another 70 years to realize a Bose–Einstein condensate in a nearly ideal gas. The main problem was to refrigerate the gas without creating a liquid or solid and without the atoms binding into molecules. To achieve this dilute, neutral gases had to be used without contact with any walls in order to avoid freezing. This presented two new challenges: a Bose–Einstein condensate in a low-density gas requires very low temperatures, • since T ρ2/3; 0 ∝ N the gas has to be refrigerated without physical contact with the outside world. •

101 By means of two advanced cooling techniques, i.e. laser cooling and evaporative cooling (see the lecture course Quantum Mechanics 2 for more details), Eric Cornell and Carl Wieman were ﬁnally able to produce the ﬁrst gaseous Bose–Einstein condensate in 1995, consisting of about a thousand rubidium-87 atoms at a temperature of 170 nK. In the same year a similar feat was realized for ultracold dilute gases of lithium-7 atoms (Randy Hulet) and sodium-23 atoms (Wolfgang Ketterle). In the latter case, the number of condensate particles had already increased a thousandfold. In 2001 Cornell, Wieman and Ketterle were awarded the Nobel prize for this achievement.

Cornell & Wieman, JILA, 1995: transition from a “normal” gas to a condensate. Velocity distribution of the gas particles for T > T , T < T and T T . 0 0 ≪ 0

Application: isotope effects for lithium. In this experiment ultracold gases were used that consisted of fermionic 6Li-atoms and bosonic 7Li-atoms. At very low temperatures we expect that a magnetically trapped 6Li-gas has a larger spatial extent than a 7Li-gas for the same number of gas atoms, since in the fermionic case the average kinetic energy is larger. As can be observed in the figure on the right, this is indeed confirmed experimentally. R.G. Hulet et al., Rice University, 2005

102 3 Relativistic 1-particle quantum mechanics

In this chapter we will attempt to cast non-relativistic quantum mechanics into a relativistically viable form. To phrase it differently, we will try to find a relativistic analogue of the Schrödinger equation. Similar material can be found in Schwabl (Ch.5–7 and Ch.10,11), Merzbacher (Ch.24) and Bransden & Joachain (Ch.15).

The desired equation should have exactly the same form for any inertial observer (relativity principle) and should have solutions that respect the quantum mechanical postulates. As a starting point for the construction of the relativistic wave equation we will use the standard connection between plane waves of the type exp(i~k ~r iωt) and free particles with · − energy E = ~ω and momentum ~p = ~~k (de Broglie hypothesis). The non-relativistic QM followed from (slightly) modifying the classical Hamilton–Jacobi equation to obtain a wave equation, which allowed to model particle–wave duality by describing the particles as a superposition of plane waves. The correct energy and momentum of the particles could be extracted from the plane waves by means of the position-space operators ∂ E = ~ω Eˆ = i~ , ~p = ~~k ~pˆ = i~ ~ , ~r ~rˆ = ~r. → ∂t → − ▽ → Applied to the classical expression E = (~r, ~p )= ~p 2/(2m)+ V (~r ) for the energy of a Hcl particle with mass m in a potential field V , the Schrödinger equation ∂ ~2 2 i~ ψ(~r, t) = (~r,ˆ ~pˆ)ψ(~r, t) = ~ + V (~r ) ψ(~r, t) ∂t Hcl − 2m ▽ is obtained. However, in this equation ~r and t are not treated on equal footing, invalidating the relativity principle. The plan is now to recast this idea into a relativistic form in flat spacetime. To this end we will adopt the following plan of attack to construct a relativistic QM for free particles.

(1) Particle–wave duality: construct a wave equation that incorporates an operator version of the correct relativistic relation between energy and momentum. This will imply that any solution to the wave equation should also satisfy the Klein– Gordon equation, providing a clear link to plane-wave decompositions (see 3.1)! § (2) Relativity principle and spin: make sure that the wave equation satisﬁes the relativity principle. By specifying this for spatial rotations, we will be able to determine the spin of the particles belonging to the wave equation!

(3) QM: extract a continuity equation from the wave equation and verify whether this allows a probability interpretation that conforms with the superposition principle.

(4) Check the non-relativistic limit and compare with experimental observations.

103 3.1 First attempt: the Klein–Gordon equation (1926)

First read appendix D, where the conventions and deﬁnitions of special relativity are summarized. The classical energy of a free relativistic particle with rest mass m is given by

E = m2c4 + ~p 2c2 . (253) p As a ﬁrst ansatz for a relativistic wave equation in position space, we could start from this classical expression and simply substitute E Eˆ = i~ ∂ and ~p ~pˆ = i~ ~ in → ∂t → − ▽ analogy with the non-relativistic case:

∂ 2 i~ ψ(x) = m2c4 ~2c2 ~ ψ(x) (x = position 4-vector) . (254) ▽ ∂t q − The “square-root operator” has to be defined by means of a power series (see the lecture course Quantum Mechanics 2), which effectively results in a differential operator of infinite order. This hugely complicates the discussion. More importantly, in this equation ~r and t are treated asymmetrically, making it not directly suitable as frame-independent expression . . . however, these drawbacks would be lifted if we would decide to apply the differential operators twice on the left- and right-hand side of the equation.

The wave equation: as a ﬁrst serious attempt it therefore makes sense to start from the squared expression E2 = m2c4 +~p 2c2 . Upon performing the substitution we obtain a wave equation that does treat ~r and t on equal footing: the so-called Klein–Gordon equation

∂2 2 ~2 ψ(x) = m2c4 ~2c2 ~ ψ(x) , (255) − ∂t2 − ▽ which is a second order diﬀerential equation in time. In contrast to the non-relativistic case, this implies that two boundary conditions are needed to fully determine the time evolution of the wave function. It should be noted, though, that by squaring the energy– momentum relation we have introduced a sign ambiguity for the energy. We will see that this aspect will come back to haunt us while trying to interpret the wave function.

Implementation of the relativity principle: to guarantee that the form of the Klein– Gordon equation is the same for each inertial observer, we ﬁrst introduce a manifest covariant notation in terms of covariant and contravariant vectors:

m2c2 ~2c2 ∂ ∂µ + m2c4 ψ(x) =0 + ψ(x) =0 , (256) µ ⇒ ~2 in terms of the d’Alembertian

2 (D.11) 1 ∂ 2 ∂ ∂µ ==== ~ . (257) ≡ µ c2 ∂t2 − ▽

104 1) Generic approach: as this will be a recurring theme in the subsequent discussions, we ﬁrst try to formulate the generic procedure to implement the relativity principle. To this end we build in the possibility that the wave function ψ(x) has several components (intrinsic degrees of freedom):

ψ1(x) ψ(x) = . . (258)  .  ψN (x)   The transformation characteristic of this (multi-component) wave function under Poincar´e transformations is deﬁned as

ψ′(x′) M(Λ)ψ(x) , (259) ≡ where the matrix M(Λ) denotes how the Poincar´etransformations act on the space spanned by the intrinsic degrees of freedom. As indicated earlier, this transformation characteristic should follow from the demand of having the same wave equation for each inertial observer:

if ψ(x)=0 for the linear diﬀerential operator and wave function ψ(x) in D D inertial system S, then it should follow automatically that also ′ψ′(x′)=0 D for the transformed diﬀerential operator ′ and wave function ψ′(x′) in inertial D system S′.

The linear diﬀerential operator is invariant under translations by a constant 4-vector aµ D since ∂/∂xµ = ∂/∂ (xµ + aµ). Therefore M(Λ) is independent of a as well as the coordinates x and x′. Furthermore, M(Λ) is a linear operator (matrix) in the space of intrinsic degrees of freedom, because the superposition principle should hold in both S and S′. Additionally we require the matrices M(Λ) to reﬂect the properties (group structure) of the Lorentz transformations:

1 1 M(Λ1Λ2) = M(Λ1)M(Λ2) and M(Λ1− ) = M − (Λ1) (260) for arbitrary Lorentz transformations Λ1 and Λ2 . This guarantees that in the absence of a Lorentz transformation also nothing happens in the space of intrinsic degrees of freedom. This ﬁxes the normalization of the matrices M(Λ) in such a way that a (projective) matrix representation of the Lorentz group is obtained.

2) Relativity principle and spin for the Klein–Gordon equation: next we specif- ically consider the Klein–Gordon equation. We know that the d’Alembertian and speed of light are scalar quantities, which do not change under the transition from one inertial frame to the other. Consequently also the Klein–Gordon operator is invariant under a change of inertial system: 2 2 2 2 m c′ m c ′ = ′ + = + = . DKG ~2 ~2 DKG 105 This differential operator acts on each component of ψ(x) separately without mixing the components, implying that each component should separately satisfy the Klein–Gordon equation. As such, the transformation matrix M(Λ) in the Klein–Gordon theory is proportional to the identity matrix. In order to guarantee the validity of the relativity principle it 11 1 hence suffices to demand that ψ(x) is a scalar field , i.e. ψ′(x′)= ψ(x)= ψ Λ− [x′ a] . − The Poincarétransformations therefore act exclusively on x-space, leaving the intrinsic degrees of freedom of ψ(x) untouched. The particles described by ψ(x) are spin-0 particles, as a spatial rotation merely affects the (spacetime) argument of the wave function without mixing the intrinsic components. The generator of infinitesimal spatial rotations is in this case exclusively given by the orbital angular momentum operator (see the lecture course Quantum Mechanics 2).

The non-relativistic limit: before investigating the probabilistic interpretation of the wave function, we ﬁrst check whether the relativistic theory reproduces non-relativistic QM at low velocities. To extract a non-relativistic wave function from the solution ψ(x) to the Klein–Gordon equation, the rest-mass term has to be removed from the time evolution:

ψ(x) ψ (x) exp( imc2t/~) . ≡ NR − After multiplication by exp(imc2t/~), the Klein–Gordon equation (255) yields

∂2 ∂ 2 ~2 + 2i~mc2 + m2c4 ψ (x) = m2c4 ~2c2 ~ ψ (x) − ∂t2 ∂t NR − ▽ NR ∂ ~2 ∂2 ~2 2 i~ ψ (x) ψ (x) = ~ ψ (x) . ⇒ ∂t NR − 2mc2 ∂t2 NR − 2m ▽ NR

The wave function ψNR(x) oscillates (in time) due to the non-relativistic kinetic energy E = E mc2 mc2, so that NR − ≪ ~2 ∂2 ∂ ψ (x) ~ ψ (x) . 2mc2 ∂t2 NR ≪ ∂t NR

In the non-relativistic limit the Klein–Gordon equation for ψ(x ) changes into a Schr¨odinger

equation for ψNR(x).

Problems with the Klein–Gordon equation: last but not least we have a closer look at the quantum mechanical interpretation of the solutions to the Klein–Gordon equation. For those solutions we will try to ﬁnd a continuity equation of the form ∂ρ/∂t + ~ ~j = 0, in ▽· analogy with what can be done in the non-relativistic case for solutions to the Schr¨odinger equation. This continuity equation will be interpreted as a manifestation of conservation

11A second possible solution is a so-called pseudoscalar ﬁeld, for which ψ′(x′) = det(Λ)ψ(x).

106 of probability, with ρ being the quantum mechanical probability density and ~j the quantum ﬂux. For solutions to the Klein–Gordon equation we can derive in a trivial way that

(256) (257) µ µ 0 ==== ψ∗(x) ψ(x) ψ(x) ψ∗(x) ==== ∂ ψ∗(x)∂ ψ(x) ψ(x)∂ ψ∗(x) − µ − (D.11) ∂ === ρ(x) + ~ ~j(x)=0 , (261) ⇒ ∂t ▽ · with i~ ∂ ∂ ρ(x) ψ∗(x) ψ(x) ψ(x) ψ∗(x) , ≡ 2mc2 ∂t − ∂t h i ~ i ~ ~ ~j(x) − ψ∗(x) ψ(x) ψ(x) ψ∗(x) . (262) ≡ 2m ▽ − ▽ h i The prefactor of ρ(x) is chosen here in such a way that in the non-relativistic limit we have ρ(x) ψ (x) 2 . For a proper quantum mechanical interpretation we would like to ≈| NR | interpret ρ(x) as relativistic probability density. To check this, we consider the plane-wave solutions to the Klein–Gordon equation (256):

E ψ (x) = exp( ip x/~) , with p p = p2 = m2c2 p0 = = m2c2 + ~p 2 . p − · · ⇒ c ± p (263) Based on these plane-wave solutions we ﬁnd that ρ(x) is not positive deﬁnite if p0 < 0 (see exer- E cise 22). As such, ρ(x) is not suitable as proba-

bility density. The next logical step is to simply re- gard all negative-energy solutions as being unphysical and remove them from the theory. This approach turns out to be not viable, since it will unavoidably 2 clash with the fundamental superposition principle mc (see exercise 22). Apart from the problems with the probabilistic interpretation, the negative-energy so- 0 lutions will result in a free-particle energy spectrum

2 that is unbounded from below (as sketched in the mc − ﬁgure on the right). This highly unstable scenario would imply that an inﬁnite amount of energy could be extracted from the system if an external perturbation would induce a transition from a positive-energy eigenstate to a negative-energy eigenstate.

107 Two strategies present themselves at this point. On the one hand we could try to interpret the negative-energy states in a different way. We will come back to this option in chapter 5, when trying to link the Klein–Gordon theory to the quantum theory of the electromagnetic field that is formulated in chapter 4. On the other hand we could follow the rather logical strategy of trying to construct an alternative relativistic wave equation that explicitly does not involve squaring the energy–momentum relation. After all, the sign ambiguity for the energy was caused by this squaring action. In 3.2 we will address § this latter (historical) approach, resulting in an entirely different wave equation that does allow for a viable probability interpretation.

Static, radially symmetric solutions to the Klein–Gordon equation: to round things off, we briefly discuss the physical applicability of the Klein–Gordon equation. To this end we consider time-independent (static) solutions φ(r) with radial symmetry. In that case the Klein–Gordon equation simplifies to

2 m2c2 1 d2 m2c2 ~ φ(r) = rφ(r) = 0 . ▽ − ~2 r dr2 − ~2 This diﬀerential equation has exponential solutions, with the physically interesting branch consisting of exponentially decreasing functions of the form

exp( r/a) ~ φYu(r) = C − , with a > 0 . (264) r ≡ mc Solutions of this type are used to describe interactions that are transmitted by massive intermediary bosons, such as the weak nuclear forces (transmitted by the W and Z bosons) and the interactions among nucleons (transmitted by pions), see the lecture course “Struc- tuur der Materie: Subatomaire Fysica”. In the latter case φYu(r) represents an attractive interaction potential of the Yukawa-type (i.e. with C < 0), where the constant length scale a = ~/mc is a measure for the eﬀective range of the potential.

If the mass would have been absent (i.e. for m = 0) we would have been dealing with the well-known Laplace equation

2 1 d2 ~ φ(r) = rφ(r) = 0 , ▽ r dr2 with solutions A φ(r) = + B. r For B =0 such a solution can be used in the description of inﬁnite-range interactions that are transmitted by massless intermediary bosons, such as the electromagnetic Coulomb- interactions (transmitted by photons) and the strong nuclear forces (transmitted by gluons).

108 3.2 Second attempt: the Dirac equation (1928)

Based on the previous observations we now use the non-relativistic expression ∂ i~ ψ(x) = Hˆ ψ(x) (265) ∂t

as starting point, with Hˆ independent of ∂/∂t. In this way the problematic differential operator ~2 ∂2/∂t2 is absent. The essential difference with the usual non-relativistic − Schrödinger equation resides in the requirement that space and time coordinates are to be treated on equal footing. As a result, Hˆ should now be taken linear in ~ . Intrinsic ▽ degrees of freedom are accounted for by writing the wave function as a column vector with N components:

ψ1(x) ψ(x) = . . (266)  .  ψN (x)   For a free particle Hˆ will not depend on t or ~r. The simplest form of the Hamilton operator then amounts to

3 Hˆ = c ~α ~pˆ + β mc2 = c αjpˆj + β mc2 . (267) · j =1 X The operators ~α and β are independent of t, ~r, E,ˆ ~pˆ and act as N N matrices on × the N components of ψ(x), bearing in mind that Hˆ should be a linear operator in order to guarantee the validity of the superposition principle. In this way we have obtained the so-called Dirac equation for a free particle with rest mass m:

∂ i~ + i~c ~α ~ β mc2 ψ(x) =0 . (268) ∂t · ▽ − In the Dirac equation the correct relativistic relation between energy and momentum should be contained. Or to phrase it diﬀerently, the matrices ~α and β should be chosen in such a way that each component of a solution to the Dirac equation should automatically be a solution to the Klein–Gordon equation.

This gives rise to the following set of matrix identities:

2 j k jk j β = IN , α ,α = 2δ IN and α , β =0 (j,k =1, 2, 3) . (269) From the anticommutation relations αj, β = 0 we can readily deduce that ~α and β should indeed be matrices. The symbol IN is used to denote an identity matrix of rank N .

109 (268) ∂ ∂ Proof: 0 ==== i~ i~c ~α ~ + β mc2 i~ + i~c ~α ~ β mc2 ψ(x) ∂t − · ▽ ∂t · ▽ − ∂2 ~2c2 3 3 = ~2 + αj,αk j k + i~mc3 αj, β j β2m2c4 ψ(x) . − ∂t2 2 ▽ ▽ ▽ − j,k =1 j =1 X X This yields the Klein–Gordon equation (255) if the matrix relations listed above are sat- isﬁed. In order to guarantee conservation of probability, we additionally have to impose that Hˆ = Hˆ †, which translates into the matrices ~α and β having to be hermitian:

j j α = (α )† and β = β† . (270)

Armed with this set of conditions we now try to ﬁnd explicit solutions for ~α and β with the lowest possible rank N.

1. The matrices ~α and β have eigenvalues 1. ± Proof: the eigenvalues of ~α and β are real, since the matrices are hermitian. More- 2 j 2 over, from the matrix relations (269) it follows that β = (α ) = IN , causing the square of the eigenvalues to be 1.

2. The matrices ~α and β are traceless. Proof: from the matrix relations in equation (269) it can for instance be inferred that cyclic αj ,β =0 Tr(αj) = Tr(β2αj) ==== Tr(βαjβ) ======{ } Tr(αjβ2) = Tr(αj) = 0. − − 3. Since the trace equals the sum of the eigenvalues, the results of steps 1 and 2 can be combined into the requirement that the matrices ~α and β should have equal numbers of +1 and 1 eigenvalues. The rank N thus has to be even. − 4. Based on steps 1 to 3, the lowest possible value for the rank is N = 2. In that case we are dealing with 2 2 matrices, spanned by the identity matrix I and the three Pauli × 2 spin matrices ~σ . However, we are looking for four diﬀerent, mutually anticommuting matrices ~α and β . This is impossible for a representation of the Dirac equation with N =2 , since the identity matrix commutes with all 2 2 matrices. × From this we can draw the conclusion that the representation of the Dirac equation with the lowest rank involves N = 4. The corresponding wave functions with four intrinsic degrees of freedom are called Dirac spinors:

ψ1(x) ψ(x) = . . (271)  .  ψ4(x)   In fact there are inﬁnitely many, physically equivalent representations for the matrices ~α and β . After all, matrix relations are invariant under arbitrary unitary transformations.

110 In the remainder of this lecture course we will adopt the so-called

I Ø Ø ~σ Dirac representation : β = 2 , ~α = , (272) Ø I ~σ Ø − 2 ! ! where the 4 4 matrices have been subdivided into 2 2 blocks. As will be proven in the × × exercise class, these matrices indeed satisfy the matrix relations (269) and (270). The corresponding proof will be based on the fundamental properties of the Pauli spin matrices 1 2 3 σ = σx , σ = σy and σ = σz . In this notation the commutator algebra for the Pauli

spin matrices reads 3 σj, σk = 2i ǫjkl σl , (273) l =1 X in terms of the completely antisymmetric coeﬃcient

+1 if (j,k,l) is an even permutation of (1, 2, 3)

ǫjkl =  1 if (j,k,l) is an odd permutation of (1, 2, 3) . (274)  −  0 else   3.2.1 The probability interpretation of the Dirac equation

As the Dirac equation is linear in both the spatial gradient and the time derivative, the probability interpretation of the Dirac equation can be obtained in a straightforward way. Deﬁne the hermitian conjugate row vector

ψ†(x) ψ∗(x), , ψ∗(x) . (275) ≡ 1 ··· 4 Then ∂ ∂ψ(x) ∂ψ†(x) (268) ~ imc ψ†(x)ψ(x) = ψ†(x) + ψ(x) ==== c ψ†(x) ~α + β ψ(x) ∂t ∂t ∂t − · ▽ ~ 2 ~ imc (270) ~ c ψ†(x) ~α †ψ(x) + ψ†(x)β†ψ(x) ==== ψ†(x)c~αψ(x) , − ▽ · ~ − ▽ · from which the following continuity equation can be extracted: ∂ ρ(x) + ~ ~j(x) =0 , (276) ∂t ▽ · with 4 2 ρ(x) ψ†(x)ψ(x) = ψ (x) and ~j(x) ψ†(x)c~αψ(x) . (277) ≡ | i | ≡ i=1 X This looks really promising, since this time ρ(x) is guaranteed to be positive deﬁnite and therefore allows an interpretation as probability density. In that interpretation the vector ~j will play the role of quantum ﬂux, which would imply that the matrix c~α can be interpreted as a kind of velocity operator in Dirac spinor space. For a viable probability interpretation it was essential that the matrices ~α and β (and as such also Hˆ ) are hermitian.

111 3.2.2 Covariant formulation of the Dirac equation

We need one more ingredient before being able to use the Dirac equation as a relativistic wave equation. We have to demand that ψ(x) transforms in such a way under Poincar´e transformations that the Dirac equation has the same form for any inertial observer. As a ﬁrst preparatory step we rewrite the Dirac equation in a form that facilitates the transition to other inertial frames. To this end we combine the four matrices β and ~α in the 4-vector- like object

γ1 γµ : γ0 β and γ2 ~γ β ~α (278) ≡   ≡ ≡ γ3  

I Ø Ø ~σ Dirac representation : γ0 = 2 , ~γ = . ⇒ Ø I ~σ Ø − 2 ! − !

These are the so-called γ-matrices of Dirac. This notation is slightly misleading in the sense that it gives the impression that γµ is a contravariant vector quantity. For that to be true an appropriate contraction in spinor space is required, which also includes the Dirac spinors in the vector quantity. The γ-matrices have the following properties (which are derived in the exercise class):

γ0 if µ =0 µ ν µν µ 0 µ 0 γ ,γ = 2g I4 (Cliﬀord algebra) , (γ )† = γ γ γ =  γj if µ = j  −  Tr(γµ)=0 , (γ0)2 = I , (γj)2 = I (j =1, 2, 3) . (279) ⇒ 4 − 4

The Dirac equation can now be rewritten in a covariant form:

β ∂ (269),(D.11) i~ + i~c ~α ~ β mc2 ψ(x) =0 ======i~γµ∂ mc ψ(x) =0 . (280) c ∂t ·▽ − ⇒ µ − µ Expressions of the type γ aµ are represented in Feynman slash notation as

exercise 23 a/ γµa = γ aµ = γ0a0 ~γ ~a ======a/ 2 = a2 I . (281) ≡ µ µ − · ⇒ 4 In this notation the Dirac equation obtains its ﬁnal compact form

i~ ∂/ mc ψ(x)=0 . (282) − In exercise 23 it will eventually be shown that the Klein–Gordon operator ( + m2c2/~2 ) can be derived straightforwardly from the Dirac operator (i~∂/ mc). − 112 Basis of 4 4 matrices: as a last preparation we introduce an explicit basis of 4 4 ma- × × trices: i I , γµ , σµν γµ,γν , γµγ5 , γ5 iγ0γ1γ2γ3 , (283) 4 ≡ 2 ≡ that has the following form in the Dirac representation (for j,k =1, 2, 3):

j I2 Ø 0 I2 Ø j Ø σ I4 = , γ = , γ = , Ø I Ø I σj Ø 2 ! − 2 ! − !

i σj, σk Ø 3 σl Ø Ø iσj σjk = − 2 = ǫjkl , σ0j = , Ø i σj, σk Ø σl iσj Ø 2 ! l =1 ! ! − X Ø I Ø I σj Ø γ5 = 2 , γ0γ5 = 2 , γjγ5 = . (284) I Ø I Ø Ø σj 2 ! − 2 ! − ! The reason why these matrices form a basis of 4 4 matrices is a trivial consequence of × the fact that each 2 2 subblock has the matrices I and ~σ as basis. Note that the ﬁve × 2 groups of matrices in (283) contain an increasing number of γ-matrices, ranging from zero to four.

3.2.3 Dirac spinors and Poincar´etransformations

The transformation characteristic of the Dirac spinors under Poincar´etransformations is deﬁned as follows [see equation (259)]:

ψ′(x′) S(Λ)ψ(x) , (285) ≡ where S(Λ) denotes how the Poincar´etransformations act on the space spanned by the intrinsic degrees of freedom (i.e. spinor space). This transformation characteristic should follow from the demand that the Dirac equation has the same form for each inertial observer: if (i~∂/ mc)ψ(x) = 0 in inertial system S, then it should follow automatically that − also (i~∂/ ′ mc)ψ′(x′) = 0 in inertial system S′. The Dirac operator (i~∂/ mc) is invari- − − ant under constant translations, since ∂/∂xµ = ∂/∂ (xµ + aµ) for a constant 4-vector aµ.

Therefore S(Λ) is independent of the coordinates x and x′. Furthermore, S(Λ) is a linear operator (matrix) in spinor space, because the superposition principle should hold in both S and S′.

If the Dirac equation (i~∂/ mc)ψ(x) = 0 should automatically imply that − (285),(D.11) µ 1 ν i~∂/ ′ mc ψ′(x′) ======i~ γ ∂ (Λ− ) mc S(Λ)ψ(x) − ν µ − Dirac eqn. µ 1 ν ν ======i~ γ (Λ− ) S(Λ) S(Λ)γ ∂ ψ(x)=0 , µ − ν 113 then S(Λ) should satisfy

ρ 1 µ 1 ν ν Λ ν 1 ρ ρ ν S− (Λ)γ (Λ− ) S(Λ) = γ ===∗ S− (Λ)γ S(Λ) = Λ γ . (286) µ ⇒ ν Next we try to ﬁnd a matrix S(Λ) for each Lorentz transformation Λ such that

1 1 S(Λ1Λ2) = S(Λ1)S(Λ2) and S(Λ1− ) = S− (Λ1) (287) for arbitrary Lorentz transformations Λ1,2 . In the absence of a Lorentz transformation accordingly nothing happens in spinor space. This ﬁxes the normalization of the matrices S(Λ) in such a way that a projective matrix representation of the Lorentz group is obtained.

(I) Rotations and boosts: we start with the proper orthochronous Lorentz transformations. Since these Lorentz transformations (and corresponding spinor transformations) can be connected continuously with the unit transformation, equation (286) only needs to be solved for inﬁnitesimal Lorentz transformations:

Λµν = gµν + δωµν , δωµν = δωνµ IR inﬁnitesimal . (288) − ∈ The reason why the tensor δωµν should be antisymmetric follows directly from the expression for the inverse of an inﬁnitesimal Lorentz transformation:

1 µν (288) µν µν (D.8) νµ (288) µν νµ (Λ− ) ==== g δω ==== Λ ==== g + δω . − Writing the corresponding transformation matrix S(Λ) in inﬁnitesimal form:

−1 −1 1 S (Λ)= S(Λ ) S(Λ) I + s(δω) S− (Λ) I s(δω) ======I + s( δω) , (289) ≈ 4 ⇒ ≈ 4 − 4 − we can derive from equation (286) that

ρ ρ ρ ν first order ρ ρ ν I4 s(δω) γ I4 + s(δω) γ + δω νγ ======γ ,s(δω) = δω νγ . − ≈ ⇒ρ ∀ (290) The transformation matrix S(Λ) can now be decomposed in terms of the basis (283) of 4 4 matrices. In exercise 24 it will be proven that all requirements are fulfilled if × s(δω)= i δω σµν. For an infinitesimal Lorentz transformation the corresponding spinor − 4 µν transformation is then given by i S(Λ) I δω σµν , δω = δω IR infinitesimal (291) ≈ 4 − 4 µν µν − νµ ∈ the matrices σµν generate the Lorentz transformations in Dirac spinor space . ⇒ For a finite Lorentz transformation of this type we get

N i ωµν µν i µν S(Λ) = lim I4 σ = exp ωµν σ , N − 4 N − 4 →∞

where the finite real coefficients ωµν are determined by the Lorentz transformation. 114 The spin operator and intrinsic spin: under a spatial rotation by an angle α about the axis oriented along the unit vector ~en , the Dirac spinors transform as (see exercise 25): ˆ ˆ ~ ~σ Ø ψ′(x′) = exp iα~en S/~ ~ ψ(x) , S~ = . (292) − · 2 Ø ~σ ! ˆ The hermitian operator S~ can be interpreted as the Dirac spin operator, since it generates the rotations in the space spanned by the four intrinsic degrees of freedom of the Dirac spinors. Because this spin operator satisfies

~2 2 ~ˆ 2 ~σ Ø (B.4) 3 ~2 S ψ(x) = 2 ψ(x) ==== ψ(x) , (293) ψ∀ 4 Ø ~σ ! 4 we are led to conclude that the Dirac equation is a relativistic wave equation for spin-1/2 particles. In chapter 5 it will be explained why the number of degrees of freedom is twice as large as expected from the non-relativistic case, where the spin space for spin-1/2 particles is 2-dimensional.

Lorentz boosts, without proof (no exam material): consider an inertial system S′ moving at constant velocity v in the ~en-direction with respect to inertial system S. In spinor space the corresponding Lorentz boost gives rise to the transformation

Ø ~σ ~en 1 1+ v/c S(Λ) = cosh(η/2) I4 sinh(η/2) · with η ln . − ~σ ~e Ø ≡ 2 1 v/c · n ! − (294) 1 This transformation is not unitary, since S†(Λ) = S(Λ) = S− (Λ). This implies that 6 the normalization of the wave function in a given spacetime point is not conserved. This change in normalization is needed to compensate for the Lorentz contraction caused by the boost (see 3.2.4). § (II) Spatial inversion and time reversal: the transformation characteristic for the discrete homogeneous Lorentz transformations can be derived straight from equation (286) by again decomposing S(Λ) in terms of the basis (283). For the parity transformation it can be deduced that (see extra exercise)

0 ψ′(x′) = S(ΛP)ψ(x) , S(ΛP) = exp(iϕP)γ (ϕP IR) . (295) ∈ Similarly we ﬁnd for the time reversal transformation that

13 ψ′(x′) = S(ΛT)ψ∗(x) , S(ΛT) = exp(iϕT)σ (ϕT IR) . (296) ∈

(III) Spacetime translations: ﬁnally we consider constant spacetime translations of the µ µ µ µ origin, i.e. x′ = x +a with a a constant 4-vector. Since the Dirac operator (i~∂/ mc) − is invariant under such translations, there is no need for a compensatory transformation in spinor space.

115 3.2.4 Solutions to the Dirac equation

The Dirac equation for a free particle involves a time-independent Hamilton operator that does not depend on ~x, i.e. H,ˆ ~pˆ = 0. As such, solutions can be found that are ˆ ˆ simultaneous eigenfunctions of Hand ~p : ψ (x) = u(~p ) exp( ip x/~) , (297) p − · where the eigenvalues E of Eˆ = i~ ∂/∂t and ~p of ~pˆ = i~~ are combined into the − ▽ contravariant momentum vector pµ with p0 = E/c. These stationary plane-wave solutions to the Dirac equation have to satisfy the additional equation

(297) i~∂/ mc ψ (x)= 0 === ( p/ mc)u(~p )=0 . (298) − p ⇒ − Solutions in the rest system (zero-momentum solutions): we start by trying to find the solutions in the so-called rest system, for which ~p = ~0 . In that case u(~0 ) should I Ø satisfy the eigenvalue equation (Eγ0 mc2)u(~0 ) = 0, where the matrix γ0 = 2 − Ø I2 − has eigenvalues 1. The desired plane-wave solutions thus read ± 1 0 E = E = mc2 : eigenvectors u(1)(~0) = 0 , u(2)(~0) = 1 , +  0   0  0 0         (299) 0 0 E = E = mc2 : eigenvectors u(3)(~0) = 0 , u(4)(~0) = 0 . − −  1   0  0 1         Negative energies and the Dirac sea: we observe again that negative-energy states occur. This time this is caused by the fact that Hˆ is linear in ~pˆ rather than quadratic. The Dirac spinors seem to describe two types of spin-1/2 particles simultaneously, one type having positive energies and one having negative energies. Again the free-particle energy spectrum is unbounded from below. However, since we are dealing with fermions here, induced transitions between positive- and negative-energy states can be avoided (Pauli-blocked). To this end Dirac assumed that all 1-fermion negative-energy states are occupied in the vacuum state of the universe. In this way the vacuum consists of an infinite sea of particles with negative energies (Dirac sea) and no particles with positive energies. The exclusion principle then forbids the unwanted transitions to these occupied negative- energy states. In 3.2.6 we will come back to the implications of this specific assumption § and we will also have a critical look at potential problems linked to the existence of such a Dirac sea.

116 Finite-momentum solutions: for completeness we have a brief look at how the plane- wave solutions change for ﬁnite momenta. These solutions can be obtained by transforming u(~0) to the correct inertial frame by means of an appropriate Lorentz boost S(Λ). How- ever, it is much simpler to make use of the fact that any solution to the Dirac equation is automatically a solution to the Klein–Gordon equation, which causes the desired plane- wave solutions to satisfy the condition p2 = m2c2. The corresponding eigenvectors can thus be obtained straight from the solutions (299) in the rest system:

2 4 2 2 E+ = m c + ~p c for r = 1, 2 u(r)(~p ) = N (p/ + mc)u(r)(~0) , E = ,  E = p m2c4 + ~p 2c2 for r = 3, 4  − −  p since (p/ mc)u(r)(~p )= N (p2 m2c2) u(r)(~0)=0 for p2 = m2c2. − − While determining the normalization factor N we should properly take into account that the normalization of ψ changes under Lorentz boosts. This is a direct consequence of conservation of probability: the probability to observe the system as being localized in a spatial volume d~x around ~x in the rest frame S should be identical to the probability to observe the system as being localized in the transformed spatial volume d~x ′ around ~x ′ in the boosted inertial frame S′. Phrased diﬀerently, we have to demand that 2 2 2 2 2 ψ(x) d~x = ψ′(x′) d~x ′, with d~x ′ = (mc /E ) d~x = d~x 1 ~v /c as a result of the | | | | + − Lorentz contraction in the direction of the boost. Consequently ψ(x) 2 should transform p | | as the time component of a contravariant vector ﬁeld, as expected from the fact that ψ(x) 2 = j0(x)/c. This translates into the normalization condition | |

(r) (s) E+ 1 u (~p )† u (~p ) δrs N = . (300) 2 2 ≡ mc ⇒ | | 2m(E+ + mc ) p Example: consider ~p = p~e . Then p/ + mc = p0 γ0 pγ3 + mc I , so that z −

2 E+ + mc 0 2 (1) N 0 (2) N E+ + mc u (p~ez) =   , u (p~ez) =   , c cp c 0    0   cp     −     

cp 0 − cp (3) N  0  (4) N   u (p~ez) = 2 , u (p~ez) = . c E+ + mc c 0    2   0   E+ + mc         

117 Remark: the twofold degeneracy of the energy eigenvalues for a given momentum eigenvalue ~p can be lifted by also specifying an appropriate spin quantum number. Such an appropriate spin quantum number is given by the spin in the direction of motion (helicity), ˆ with corresponding operator Sˆ(~p ) = S~ ~e (see exercise 26). In the example discussed · p above, u(r)(p~e ) has helicity ~/2 if r = 1, 3 and ~/2 if r = 2, 4. The fact that z − H,ˆ ~pˆ and Sˆ(~p ) are mutually commuting operators follows from a symmetry argument that is valid for arbitrary relativistic free-particle theories. First of all, a free-particle theory is invariant under translations, since empty space is completely isotropic. This implies that Hˆ and ~pˆ commute. The same in principle holds for rotational invariance. But mind you, ~pˆ actually introduces an explicit direction. As a result, rotational symmetry is restricted to rotations about this direction, and Hˆ and ~pˆ exclusively commute with the ˆ ˆ angular momentum component ~pˆ J~ = ~pˆ S~ . · ·

3.2.5 The success story of the Dirac theory

At this point we should have a closer look at the implications of the Dirac theory. We start with its successes. To this end we assume that we are dealing with spin-1/2 particles with charge q and rest mass m. Subsequently we consider the interaction with classical electromagnetic ﬁelds described by a real vector potential A~(x) and a real scalar potential φ(x). The associated interaction can be obtained in the usual way through minimal substitution:

Eˆ Eˆ qφ(ˆx) and ~pˆ ~pˆ qA~(ˆx) . (301) → − → − In terms of the electromagnetic 4-vector potential

A1(x) Aµ(x): A0(x) φ(x)/c and A2(x) A~(x) (302) ≡   ≡ A3(x)   minimal substitution can be written in covariant form:

position repr. pˆµ pˆµ qAµ(ˆx) ======i~∂µ i~∂µ qAµ(x) . (303) → − ⇒ → −

In this way we obtain the Dirac equation for spin-1/2 particles with charge q and rest mass m that are subjected to a classical electromagnetic ﬁeld:

i~∂/ mc qA/(x) ψ(x) =0 . (304) − − To see what has changed with respect to the non-relativistic case, three special scenarios will be considered of which only the ﬁrst one will be worked out in full detail.

118 (I) Non-relativistic limit: intrinsic magnetic dipole moment. As a first example we consider the interaction with static electromagnetic fields, such as constant magnetic fields or the Coulomb potential generated by an atomic nucleus. A static electromagnetic field is described classically by a time-independent vector potential A~(~x ) and a time-independent scalar potential φ(~x ). The corresponding Hamilton operator

position repr. Hˆ ======c~α i~~ qA~(~r ) + qφ(~r )+ β mc2 (305) · − ▽ − does not depend on t x0/c but it does depend on ~r ~x . Therefore it makes sense to ≡ ≡ look for stationary solutions of the type ∂ ψ(~r, t) = ψ(~r ) exp( iEt/~) , Hˆ ψ(~r, t) = i~ ψ(~r, t) = E ψ(~r, t) . (306) − ∂t

2 Next we select a positive-energy solution, we write E = ENR + mc in order to split oﬀ the rest-mass term, and we decompose the Dirac spinor in terms of 2-dimensional spinors

ψU and ψD : ψ (~r ) ψ(~r ) U . (307) ≡ ψD (~r ) !

From the eigenvalue equation Hˆ ψ(~r, t)= E ψ(~r, t) we obtain in the Dirac representation [see equation (272)]

(E + mc2)ψ = (qφ + mc2)ψ + c~σ ( i~~ qA~ )ψ , NR U U · − ▽ − D (E + mc2)ψ = c~σ ( i~~ qA~ )ψ + (qφ mc2)ψ . (308) NR D · − ▽ − U − D The mc2 terms drop out from the ﬁrst equation, but not from the second equation. As a result we obtain

c~σ ( i~~ qA~ )ψ i~~ qA~ ▽ U NR limit ▽ ψD = · − −2 ======ψD ~σ − − ψU , (309) ENR +2mc qφ ⇒ ≈ · 2mc − bearing in mind that in the non-relativistic limit the total and potential non-relativistic energies are substantially smaller than the rest-mass energy, i.e. E , qφ mc2. | NR| | | ≪ This means that the 2-dimensional (“small”) component ψD is suppressed by a factor (p/mc) = (v/c) with respect to the other 2-dimensional (“large”) component ψ . O O U Inserting the limit (309) in the ﬁrst equation of (308) yields

1 2 E ψ = qφ ψ + ~σ ( i~~ qA~ ) ψ NR U U 2m · − ▽ − U h i 1 i = qφ + ( i~~ qA~ )2 + ~σ ( i~~ qA~ ) ( i~~ qA~ ) ψ , 2m − ▽ − 2m · − ▽ − × − ▽ − U h i

119 which gives rise to the so-called Pauli equation for the 2-dimensional spinor ψU :

1 q~ E ψ = qφ + ( i~~ qA~ )2 ~σ ~ ψ . (310) NR U 2m − ▽ − − 2m · B U h i To arrive at this expression we made use of the operator identity (B.6) and the relation

( i~~ qA~ ) ( i~~ qA~ ) = i~q(~ A~ ) = i~q ~ . − ▽ − × − ▽ − ▽ × B The first two terms on the right-hand side of the Pauli equation we recognize as the interaction between a charged spin-0 particle and a classical electromagnetic field. The last term is one of the big successes of the Dirac theory. The Dirac theory predicts the experimentally observed interaction between the magnetic field ~ and the spin-induced ˆ B intrinsic magnetic dipole moment ~ : MS

ˆ q~ e ~ Hˆ spin = ~ ~ = ~σ ~ | | g ~σ ~ . (311) B − MS · B − 2m · B ≡ 4m · B

Consequence: an elementary particle with charge q = e that is described by the Dirac −| | theory (such as an electron or muon) automatically has a lowest-order g-factor of 2. This g-factor agrees with experimental observations and could not be explained in the non- relativistic theory! It should be noted though that non-elementary particles (such as antiprotons), which can be considered as consisting of even smaller particles, in general do not require g = 2 at lowest order.

(II) Relativistic corrections: ﬁne structure for central potentials. Consider an electrostatic central potential, for which A~(~r ) = ~0 and qφ(~r ) = V (r). By taking the afore-mentioned non-relativistic approximation one step further, we can obtain the ﬁrst-order relativistic corrections to the Pauli equation. This results in three relativistic corrections to the non-relativistic Hamilton operator (without proof):

~pˆ 4 ~2 ~ ˆ ∆Hˆ = + (~rˆ) + (ˆr) L~ ~σ, FS − 8m3c2 8m2c2 V2 4m2c2 V3 · 2 1 dV (r) with (~r ) ~ V (r) and (r) , (312) V2 ≡ ▽ V3 ≡ r dr which coincides with the three perturbative corrections that are for instance responsible for the fine structure of 1-electron atoms (see the lecture course Quantum Mechanics 1). The first term is a relativistic correction to the kinetic energy. The second term is a relativistic correction to the potential energy (called Darwin term). The last term is the well-known spin–orbit interaction, which describes the interaction between the spin-induced intrinsic magnetic dipole moment of the spin-1/2 particle and the magnetic field that is caused by the orbital motion (orbital angular momentum) of the particle.

120 (III) Discrete energy levels of the relativistic 1-electron atom. Again consider the Hamilton operator for an electrostatic central potential

Hˆ = c~α ~pˆ + β mc2 + V (ˆr) . (313) ·

ˆ 2 ˆ 2 In the non-relativistic case the operators H,ˆ L~ , Lˆz , S~ , Sˆz and ˆ formed a complete set ˆ ˆ P of commuting observables. However, L~ and S~ no longer individually commute with the Hamilton operator due to the presence of relativistic spin interactions in the Dirac equation ˆ ˆ ˆ (such as the spin–orbit interaction). The total angular momentum operator J~ = L~ + S~ does commute with the Hamilton operator, since V (r) respects the rotational symmetry of empty space. The energy levels of the relativistic 1-electron atom can therefore be ˆ specified in terms of the quantum number j belonging to J~ 2, which is typical for the relativistic QM. The orbital angular momentum quantum number ℓ is not a “good quantum number” anymore, but the total angular momentum quantum number j is. In the absence of a preferred direction, the energy levels will be independent of the magnetic quantum number m belonging to Jˆ . Note also that the non-relativistic parity operator ˆ no j z P longer commutes with the Hamilton operator, since Hˆ is linear in ~pˆ and ˆ, ~pˆ = 0. P The relativistic parity operator β ˆ does commute with the Hamilton operator because P β, ~α = 0, with β representing the action of the parity transformation in spinor space [see S (ΛP ) in equation (295)]. For a 1-electron atom with V (r)= Zα~c/r the energy spectrum consists of two contin- − uous branches with E m c2 and a discrete set of positive-energy levels corresponding | | ≥ e to bound states (see 15.5 of Bransden & Joachain for more details): § 1/2 − 2 2 (Zα) Enj = mec 1 + 2  ,  n j 1/2 + (j +1/2)2 (Zα)2   − − −   p  with n = 1, 2, and j = 1/2, 3/2, , n 1/2 . (314) ··· ··· − These energy levels are (2j+1)-fold degenerate as a result of the magnetic quantum number m = j , , j . Moreover, each level has an additional twofold degeneracy due to the j − ··· parity quantum number ( 1)ℓ with ℓ = j 1/2, with the exception of j = n 1/2 for − ± − which only ℓ = j 1/2 = n 1 is allowed. This so-called fine structure is confirmed − − experimentally at high precision. In the non-relativistic limit one can expand in powers of v/c = (Zα/n): O 2 2 2 1 2 (Zα) (Zα) n 3 Enj mec 2 mec 2 1 + 2 + , (315) − ≈ − n n j +1/2 − 4 ··· # which agrees with the result derived in the lecture course Quantum Mechanics 1.

121 3.2.6 Problems with the 1-particle Dirac theory

As a relativistic wave equation, the Dirac equation gives rise to a few aspects that are tough to interpret. As we have seen, there is the problem with the negative-energy states and the ensuing energy spectrum that is unbounded from below. Transitions from positive- to negative-energy states can be avoided by invoking the exclusion principle and assuming that all negative-energy states are occupied. This is rather unsatisfactory from a 1-particle perspective, since the relevant number of particles is in fact inﬁnite and we actually should take into account the interactions among these particles. On top of that, an inﬁnite negative vacuum energy may not pose a problem at the level of quantum mechanics, but it may have cosmological implications.

Question: When do we expect to see observable manifestations of the presence of the Dirac sea, which would force us to let go of the Dirac theory as a 1-particle theory?

In order to excite a particle from the Dirac sea a minimum 2mc2 of energy is required. The excitation would leave behind a hole in the sea and the corresponding negative-energy particle would be promoted to a positive-energy particle. This pair creation of a particle and a hole is the relativistic analogue of the electron–hole excitation in hole theory (see 1.7.1). §

A quantum mechanical limitation to the 1-particle theory: consider a particle with positive energy that is localized in a box of size L. According to Heisenberg’s uncertainty principle, the quantum mechanical uncertainty ∆p on the momentum of the particle is in that case at least (~/L). The uncertainty on the relativistic energy of the particle O would roughly amount to c∆p = (~c/L). If this uncertainty exceeds the 2mc2 bar- O rier pair creation can occur and the Dirac sea will manifest itself. Hence, if the particle is localized in a box of size L = (~/mc), which is referred to as the Compton wavelength O of the particle, we are in fact not dealing with a 1-particle theory anymore.

Quantum mechanics thus involves two characteristic length scales for a particle with mass m and momentum p: on (~/p) length scales the quantum- O mechanical wave-like nature of the particle manifests itself and on even smaller length scales of (~/mc) we have to let go of the concept of an individual O particle.

In the latter case it makes no sense anymore to talk about the concept of conservation of probability. After all, additional particles can be created. We will therefore have to assign a diﬀerent meaning to the conservation law that was responsible for the continuity equation (276). Note: in the non-relativistic limit many-particle aspects do not play a prominent role yet. After all, for non-relativistic momenta ~p mc also ∆~p mc | | ≪ | | ≪ and therefore the uncertainty on the position of the particle is at least ∆~r (~/mc), | | ≫ O which largely exceeds the Compton wavelength.

122 4 Quantization of the electromagnetic ﬁeld

We are in a bit of a pickle now. We have no clue how to interpret the successes and shortcomings of the relativistic 1-particle quantum theories. To be able to make any conclusive statements, we turn our attention now to a theory for which we already know the correct relativistic wave equation . . . electromagnetism.

In this chapter the electromagnetic ﬁeld will be quantized by applying the usual canonical quantization conditions to the classical electromagnetic Hamiltonian. In the end we will see that the quantization of the electromagnetic ﬁeld will turn out to be a very natural application of second quantization, as described in 1.5. The free electromagnetic theory actually is a many-particle theory for § massless spin-1 quanta, called photons. Similar material can be found in Schwabl (Ch.14) and Merzbacher (Ch.23).

4.1 Classical electromagnetic ﬁelds: covariant formulation

We start from the Maxwell equations for an electromagnetic ﬁeld in vacuum with charge

density ρc(~r, t) and current density ~jc(~r, t):

∂ ~ ~(~r, t) = 0 , ~ ~(~r, t) = ~(~r, t) , (316) ▽ · B ▽ × E − ∂t B

~ 1 ~(~r, t) = ρc(~r, t) , (317) ▽ · E ǫ0

1 ∂ ~ ~(~r, t) = ~(~r, t) + µ ~j (~r, t) . (318) ▽ × B c2 ∂t E 0 c

Here ~(~r, t) and ~(~r, t) are the real electric and magnetic ﬁelds. In addition, the con- E B stants ǫ0 and µ0 are the permittivity and permeability of the vacuum, which are linked to the speed of light c in vacuum through the relation

2 ǫ0µ0 = 1/c . (319)

Next we introduce the real scalar potential φ′(~r, t) and the real vector potential A~ ′(~r, t), such that

~ ∂ ~ ~(~r, t) = φ′(~r, t) A~ ′(~r, t) , ~(~r, t) = A~ ′(~r, t) . (320) E − ▽ − ∂t B ▽ ×

123 In this way the ﬁrst two Maxwell equations (316) are satisﬁed automatically, since on ~ ~ ~ ~ general grounds A~ ′(~r, t) = 0 and φ′(~r, t) = ~0 . The Maxwell ▽ · ▽ × ▽ × ▽ equation (317) can be rewritten now as 1 ∂ ~ ~ ~ ρc(~r, t) = A~ ′(~r, t) φ′(~r, t) ǫ0 − ∂t ▽ · − ▽ · ▽

2 1 ∂ ~ 2 ∂ ~ 1 ∂ = φ′(~r, t) A~ ′(~r, t) + φ′(~r, t) . c2 ∂t2 − ▽ − ∂t ▽ · c2 ∂t By employing the identity

~ ~ general ~ ~ ~ 2 A~ ′(~r, t) ======A~ ′(~r, t) A~ ′(~r, t) ▽ × ▽ × ▽ ▽ · − ▽ the Maxwell equation (318) ﬁnally becomes

2 ~ ~ 1 ∂ 1 ∂ ~ µ ~j (~r, t) = A~ ′(~r, t) + A~ ′(~r, t) + φ′(~r, t) 0 c ▽ × ▽ × c2 ∂t2 c2 ∂t ▽

2 1 ∂ ~ 2 ~ ~ 1 ∂ = A~ ′(~r, t) + A~ ′(~r, t) + φ′(~r, t) . c2 ∂t2 − ▽ ▽ ▽ · c2 ∂t Next we deﬁne the electromagnetic 4-vector potential

1 A′ (x) µ 0 2 A′ (x): A′ (x) φ′(~r, t)/c and A′ (x) A~′ (~r, t) (321) ≡  3  ≡ A′ (x)   and the electromagnetic 4-current

1 jc (x) µ 0 2 ~ jc (x): jc (x) cρc(~r, t) and jc (x) jc(~r, t) . (322) ≡  3  ≡ jc (x)   2 By inserting the relation 1/ǫ0 = µ0c the Maxwell equations take their final covariant form µ µ ν µ A′ (x) ∂ ∂ A′ (x) = µ j (x) . (323) − ν 0 c By taking the 4-divergence of this wave equation, we readily derive that the electromagnetic µ 4-current has to satisfy the continuity equation ∂µ jc (x) = 0. This reflects the conservation of total electric charge, as opposed to the quantum mechanical probability interpretation that we would have assigned to it in the previous chapter. As a result of the relativity µ principle, A′ (x) should transform as a contravariant vector field (see exercise 28). If regarded as a quantum mechanical wave function, the electromagnetic 4-vector potential therefore should describe spin-1 particles.

124 µ For the quantum mechanical interpretation of A′ (x) we restrict the discussion to a free electromagnetic field without charges or charge currents. In that case the wave equation reduces to µ µ ν A′ (x) ∂ ∂ A′ (x) = 0 . (324) − ν Note that the potentials φ′(~r, t) and A~ ′(~r, t) are not fixed completely by the condition (320). For arbitrary real differentiable functions χ(~r, t), the transformed potentials

~ ∂ A~(~r, t) = A~ ′(~r, t) χ(~r, t) and φ(~r, t) = φ′(~r, t) + χ(~r, t) ▽ − ∂t (325) µ µ µ A (x) = A′ (x) + ∂ χ(x) ⇒

give rise to exactly the same electric/magnetic ﬁelds and therefore describe the same physics. The associated freedom to choose the potentials is referred to as gauge freedom. This gauge freedom can be exploited to transform the potentials in such a way that the Coulomb conditions

~ A~(~r, t) = 0 and φ(~r, t)=0 , (326) ▽ · are satisﬁed, which implies that

∂ ~(~r, t) = A~(~r, t) and ~(~r, t) = ~ A~(~r, t) . (327) E − ∂t B ▽ ×

This choice of gauge is called Coulomb gauge. We prefer this non-covariant gauge con- µ dition to a covariant condition such as ∂µA (x) = 0, since the former allows us to more easily identify the two physical degrees of freedom of the theory. In a covariant gauge the two remaining, unphysical (read: physically non-relevant) degrees of freedom have to be taken into account as well. In the Coulomb gauge the Maxwell equations simplify to the free electromagnetic wave equation

1 ∂2 2 A~(~r, t) = ~ A~(~r, t)=0 , ~ A~(~r, t)=0 , (328) c2 ∂t2 − ▽ ▽ · implying that each component of A~(~r, t) satisﬁes the Klein–Gordon equation with m =0. Also the electric and magnetic ﬁelds satisfy this simple massless Klein–Gordon equation.

4.1.1 Solutions to the free electromagnetic wave equation

Next we will solve the wave equation (328) for A~(~r, t) inside a ﬁnite cubic enclosure with edges L and periodic boundary conditions. These restrictions can be lifted later on by taking the continuum limit L (see App. A.2). The periodicity condition allows us to →∞

125 express the solutions in terms of plane waves with quantized values for the wave vectors (propagation vectors) ~k = ~p/~ featuring in the spatial Fourier modes exp(i~k ~r ): ∝ · ωk 2.5.2 2π ~k k ~e ~e ====§ ~ν, with ν = 0, 1, 2, ~ν = ~0 . (329) ≡ k ≡ c k L x,y,z ± ± ··· ∧ 6 The Fourier mode corresponding to ~ν = ~0 is left out here, since it does not refer to an electromagnetic wave. The general solution to the free electromagnetic wave equation then reads

positive frequency sol. negative frequency sol. 1 2 ~ ~ ~ ∗ ~ A(~r, t) = ~ǫλ(~ek) a~k,λ exp(ik ~r iωkt)+ a~k,λ exp(iωkt ik ~r ) √V ǫ 2ωk · − − · 0 ~ λ=1 r Xk X h z }| { z }| { i 1 2 ~ ∗ = ~u~k,λ(~r ) a~k,λ(t) + ηλ a ~k,λ(t) √ǫ0 2ωk ~ λ=1 r − Xk X h i (330) in terms of the Fourier components Fourier mode unit vector exp(i~k ~r ) ~u~ (~r ) · ~ǫλ(~ek) . (331) k,λ ≡ z √}|V { ∗ z }| {

The factor ~/2ωk has been introduced to render the Fourier coeﬃcients

p a (t) a exp( iω t) (332) ~k,λ ≡ ~k,λ − k dimensionless, which is crucial for performing the quantization. As a result of the Coulomb condition (326), one of the three degrees of freedom of the vector potential A~(~r, t) is not physical, since it can be gauged away. This aspect is manifested in the transversality of the polarization vectors ~ǫ (~e ) IR3 : λ k ∈

~ A~ = 0 ~ǫ (~e ) ~k ======▽· 0 , with ~ǫ 2(~e ) = 1 and ~ǫ (~e ) ~ǫ (~e )=0 . (333) λ k · λ k 1 k · 2 k

These polarization vectors are chosen in such a way that the unit vectors ~ǫ1(~ek) , ~ǫ2(~ek) and ~ek form a right-handed coordinate system:

~ǫ (~e ) ~ǫ (~e ) = ~e , ~ǫ (~e ) ~e = ~ǫ (~e ) , ~e ~ǫ (~e ) = ~ǫ (~e ) , 1 k × 2 k k 2 k × k 1 k k × 1 k 2 k 2 k k completeness ǫ (~e )ǫ (~e ) + j l ======δ (j,l = x,y,z ) , (334) λ,j k λ,l k k k jl λX=1 and that this right-handedness is preserved under spatial inversion ~k ~k : → − ~k ~k ~ǫ (~e ) →− ~ǫ ( ~e ) η ~ǫ (~e ) , with η = + 1 and η = 1 . (335) λ k −−−−→ λ − k ≡ λ λ k 1 2 −

126 Some more remarks about equation (330).

In the second line of equation (330) we replaced ~k by ~k in the second term. • − Subsequently we exploited the inversion symmetry of the ~k sum and used the transformation property (335) of the polarization vector. It will depend on the explicit application which of the two expressions for the vector potential will turn out to be the most useful one. For example, in order to read oﬀ that A~(~r, t) is real the ﬁrst expression is more suitable, whereas the second expression will be ideal for performing spatial integrals (as we will see).

In the previous chapter we would at this point have interpreted the positive-frequency • terms exp( iω t) as positive-energy solutions and the negative-frequency terms ∝ − k exp(iω t) as negative-energy solutions. That is precisely what we will not do in ∝ k this case, since we actually know what the classical energy of the system should be.

4.1.2 Electromagnetic waves in terms of harmonic oscillations

The next step involves determining the Hamiltonian (energy) of the free electromagnetic ﬁeld for a given solution (330). From classical electrodynamics we know that this Hamil- tonian is given by

ǫ (327) ǫ ∂A~(~r, t) 2 2 = 0 d~r ~ 2(~r, t)+ c2 ~ 2(~r, t) ==== 0 d~r + c2 ~ A~(~r, t) , H 2 E B 2 ∂t ▽× VZ h i VZ (336) where ρ (~r, t) = ǫ ~ 2(~r, t)+ c2 ~ 2(~r, t) /2 is the energy density of the classical elec- H 0 E B tromagnetic field. By means of the orthonormality relations for the Fourier components, which are given in appendix E, this energy density can be expressed in terms of the independent coefficients a (t) and a∗ (t). For the electric-field contribution to the energy ~k,λ ~k,λ density this proceeds as follows:

2 (336) ǫ0 ∂A~(~r, t) A~ IR3 ǫ0 ∂A~(~r, t) ∂A~∗(~r, t) ==== d~r ====∈ d~r HE 2 ∂t 2 ∂t · ∂t VZ VZ 2 (330) 1 ~ ∗ ∗ ′ ′ ′ ==== a˙ ~k,λ(t)+ ηλ a˙ ~k,λ(t) a˙ ~k′,λ′ (t)+ ηλ a˙ ~k ,λ (t) 2 2√ωk ωk′ − ∗ ~ ~ ′ λ,λ′=1 − Xk,k X 2 ~ (E.5) 1 2 ∗ ∗ d~r~u~k,λ(~r ) ~u~k′,λ′ (~r ) ==== a˙ ~k,λ(t)+ ηλ a˙ ~k,λ(t) ∗ · 4 ωk | − | VZ X~k λX=1 2 (332) 1 2 ~ ∗ ==== ωk a~k,λ(t) ηλ a ~k,λ(t) . 4 | − − | X~k λX=1

127 The magnetic-ﬁeld contribution to the energy density can be obtained in a similar way:

2 2 2 (336) c ǫ0 ~ A~ IR3 c ǫ0 ~ ~ ==== d~r A~(~r, t) ====∈ d~r A~(~r, t) A~∗(~r, t) HB 2 ▽× 2 ▽× · ▽× VZ VZ 2 ~ (330) 1 2 ∗ ∗ ′ ′ ′ ==== c a~k,λ(t)+ ηλ a ~k,λ(t) a~k′,λ′ (t)+ ηλ a ~k ,λ (t) 2 2√ωk ωk′ − ∗ ~ ~ ′ λ,λ′=1 − Xk,k X

~ ~ d~r k ~u (~r ) k ~u ∗′ ′ (~r ) ∗ × ~k,λ · ′ × ~k ,λ VZ 2 (E.6),(329) 1 2 ~ ∗ ======ωk a~k,λ(t)+ ηλ a ~k,λ(t) . 4 | − | X~k λX=1 This can be combined into the compact expression

2 2 1 ~ 2 2 ~ 2 = ωk a~k,λ(t) + a ~k,λ(t) = ωk a~k,λ(t) . (337) H 2 | | | − | | | X~k λX=1 X~k λX=1 Because a (t) 2 = a 2 we readily obtain that the Hamiltonian is properly time inde- | ~k,λ | | ~k,λ| pendent, as expected for an isolated system. Finally, we rewrite the result in terms of the real quantities ~ ∗ Q~k,λ(t) a~k,λ(t)+ a~k,λ(t) ≡ 2ωk h i r (332) ~ ˙ ∗ and P~k,λ(t) Q~k,λ(t) ==== iωk a~k,λ(t) a~k,λ(t) . (338) ≡ − − 2ωk h i r The Hamiltonian becomes much more accessible in this way:

1 2 = P 2 (t) + ω2 Q2 (t) = Q , P . (339) H 2 ~k,λ k ~k,λ H { ~k,λ} { ~k,λ} ~ λ=1 Xk X

The quantities Q~k,λ and P~k,λ satisfy the Hamilton equations

∂ (338) ˙ ∂ 2 (332) ¨ (338) ˙ H = P~k,λ ==== Q~k,λ and H = ωk Q~k,λ ==== Q~k,λ ==== P~k,λ . ∂P~k,λ ∂ Q~k,λ − − As such they form conjugate pairs of coordinates and momenta.

An electromagnetic wave in vacuum is the result of harmonic oscillations of the electromagnetic ﬁeld. The Hamiltonian consists of a countably inﬁnite set of independent linear

harmonic oscillators, characterized by the direction of propagation ~ek of the wave, its

polarization vector ~ǫλ(~ek) and its frequency ωk = ck.

128 4.2 Quantization and photons

Since the Hamiltonian simply corresponds to a set of linear harmonic oscillators, we can quantize along the lines of 1.6.1 and 1.5. The Hamiltonian is then promoted to a § § ˆ Hamilton operator in terms of the generalized position operators Q~k,λ(t) in the Heisenberg ˆ picture and the corresponding conjugate momentum operators P~k,λ(t). These operators satisfy the fundamental (canonical) commutation relations

ˆ ˆ ~ ′ ˆ ˆ ˆ ˆ ˆ Q~k,λ(t), P~k′,λ′ (t) = i δλ,λ δ~k,~k′ 1 , Q~k,λ(t), Q~k′,λ′ (t) = P~k,λ(t), P~k′,λ′ (t) = 0 . (340) Subsequently we deﬁne creation, annihilation and number operators in the Heisenberg picture according to

ωk i creation operators :a ˆ† (t) Qˆ~ (t) Pˆ~ (t) aˆ† exp(iωkt) , ~k,λ ≡ 2~ k,λ − √2~ω k,λ ≡ ~k,λ r k annihilation operators :a ˆ (t) aˆ exp( iω t) , ~k,λ ≡ ~k,λ − k

number operators :n ˆ~ (t) aˆ† (t)ˆa~ (t) =a ˆ† aˆ~ nˆ~ . (341) k,λ ≡ ~k,λ k,λ ~k,λ k,λ ≡ k,λ These creation and annihilation operators satisfy the bosonic commutation relations

aˆ (t), aˆ† (t) = δ ′ δ ′ 1ˆ , aˆ (t), aˆ ′ ′ (t) = aˆ† (t), aˆ† (t) = 0 . (342) ~k,λ ~k′,λ′ λ,λ ~k,~k ~k,λ ~k ,λ ~k,λ ~k′,λ′ In analogy with the Hamilton operator (76) for a single linear harmonic oscillator, we obtain the following Hamilton operator for the quantized electromagnetic ﬁeld:

2 2 1 1 Hˆ = ~ω aˆ† aˆ +a ˆ aˆ† = ~ω nˆ + 1ˆ . (343) 2 k ~k,λ ~k,λ ~k,λ ~k,λ k ~k,λ 2 ~ λ=1 ~ λ=1 Xk X Xk X Such observables belonging to collective quantities of the electromagnetic field can be obtained in an alternative way by simply replacing the fields in the classical quantities by the corresponding field operators in the Heisenberg picture. The real vector potential of equation (330) is then changed into the corresponding hermitian electromagnetic field operator

2 ˆ 1 ~ A~(~r, t) = ~ǫ (~e ) aˆ~ exp(i~k ~r iω t)+â† exp(iω t i~k ~r ) λ k k,λ k ~k,λ k √V ǫ 2ωk · − − · 0 ~ λ=1 r Xk X h i 1 2 ~ = ~u~ (~r ) aˆ~ (t) + η aˆ† (t) k,λ k,λ λ ~k,λ √ǫ0 2ωk ~ λ=1 r − Xk X h i (344) in the Heisenberg picture. This field operator satisfies the same free electromagnetic wave

129 equation as the classical vector potential. The quantized versions of the electric and magnetic ﬁelds follow from the usual relations ˆ ∂ ˆ ˆ ˆ ~ (~r, t) = A~(~r, t) and ~ (~r, t) = ~ A~(~r, t) . (345) E − ∂t B ▽ ×

Using appendix E as well as the commutation relations (342) the following can be derived (see exercise 29 for an example).

The total Hamilton operator for the quantized electromagnetic ﬁeld reads • 2 ǫ0 ˆ ˆ 2 ˆ ˆ 1 Hˆ d~r ~ (~r, t) ~ †(~r, t)+ c ~ (~r, t) ~ †(~r, t) = ~ω nˆ + 1ˆ . ≡ 2 E · E B · B k ~k,λ 2 Z ~k λ=1 V h i X X (346)

The total momentum operator for the quantized electromagnetic ﬁeld follows from • the Poynting vector and reads

2 ˆ ˆ ˆ P~ ǫ d~r ~ (~r, t) ~ †(~r, t) = ~~k nˆ . (347) ≡ 0 E × B ~k,λ VZ h i X~k λX=1

The total spin operator for the quantized electromagnetic ﬁeld reads • ˆ ˆ ˆ S~ ǫ d~r ~ (~r, t) A~ †(~r, t) = ~~e nˆ nˆ , (348) ≡ 0 E × k ~k,R − ~k,L Z ~k V h i X

where the number operatorsn ˆ~ =a ˆ† aˆ~ andn ˆ~ =a ˆ† aˆ~ are defined k,R ~k,R k,R k,L ~k,L k,L according to 1 1 aˆ~ = (â~ iaˆ~ ) anda ˆ~ = (â~ + iaˆ~ ) . (349) k,R − √2 k,1 − k,2 k,L √2 k,1 k,2

4.2.1 Particle interpretation belonging to the quantized electromagnetic ﬁeld

From the above-given observables for collective electromagnetic quantities we can directly read oﬀ the particle interpretation of the quantized free electromagnetic theory. Ow- ing to the bosonic commutation relations, the electromagnetic ﬁeld can be regarded as being composed of bosonic quanta, called photons. The operatora ˆ† creates such a ~k,λ ~~ ~ ~ photon with momentum k, energy ωk = ck and polarization λ, whereasa ˆ~k,λ an- nihilates such a photon. The number operatorn ˆ~k,λ counts such photons. From the additive spin operator (348) it can be concluded that photons are spin-1 particles, as already predicted in exercise 28 by employing the relativity principle. However, the

helicity of the photon, i.e. the spin component along the direction of propagation ~ek ,

130 can exclusively take on the values + ~ (right-handed circularly polarized light) and ~ − (left-handed circularly polarized light). The absence of the helicity eigenvalue 0 is a direct consequence of the fact that one of the three degrees of freedom of the vector ﬁeld can be gauged away and is therefore unphysical. This statement only makes sense as being independent of the frame of reference if the photons are massless, which indeed follows from the relativistic relation between mass, energy and momentum:

(329) m2c4 = E2 ~p 2c2 = ~2ω2 ~2k2c2 ==== 0 . − k − If the photons would have had a ﬁnite mass, then a rest system would exist with respect to which the photons would be at rest. In that case no direction of propagation and therefore no helicity could have been deﬁned. Massless particles move at the speed of light in any inertial frame, excluding the existence of a rest frame.

4.2.2 Photon states and density of states for photons

Based on 1.2.1, the multi-photon states can be described in terms of the bosonic eigen- § states of the number operator:

n n nj (â† ) 1 (â† ) 2 (â†) n , n , = 1 2 0 j 0 , | 1 2 ···i √ √ ··· | i ≡ | i n1! n2! j nj! . (350) Y with 0 = the normalized photon vacuump | i

Just like we did in chapter 1, the label j =1, 2, numbers the complete set of quantum ··· numbers ~k and λ. The multi-photon states have the following bosonic properties [see equation (30)]:

nˆ , n , = n , n , , j |··· j ···i j |··· j ···i

aˆj , nj, = √nj , nj 1, , |··· ···i |··· − ···i

aˆ† , n , = n +1 , n +1, . (351) j |··· j ···i j |··· j ···i p The state function of the radiation ﬁeld can be described in terms of this complete basis, with the size of a particular occupation number representing the strength of the corresponding monochromatic component for the given radiation ﬁeld.

For later use we also derive the density of states for photons with wave vector ~k and polarization λ. In complete analogy to the steps presented in 2.5.2, we determine the number § of 1-photon states with energy on the interval [E, E + dE ], polarization λ, and ~ek within

131 the solid angle dΩ around the ﬁxed (θ,φ)-direction ~eΩ :

V k = E/(~c) V E2 ρ (E, Ω) dΩ dE = k2dk dΩ ======dΩ dE λ 8π3 (2π~c)3 V E2 ρ (E, Ω) = (λ =1, 2) . (352) ⇒ λ (2π~c)3

4.2.3 Zero-point energy and the Casimir eﬀect

For the total Hamilton operator in equation (346) we have

(346) Hˆ n , n , ==== ~ω n + E n , n , , (353) | 1 2 ···i j j 0 | 1 2 ···i j X where ~ωj is the energy of the photon mode labeled by j . As in the case of the linear harmonic oscillator, here too we encounter a zero-point energy E0 :

1 2 Hˆ 0 = E 0 = ~ω 0 = ~ω 0 , (354) | i 0 | i 2 k | i k | i X~k λX=1 X~k 1 which originates from the operator combinations (â† aˆ~ +a ˆ~ aˆ† ). The zero-point 2 ~k,λ k,λ k,λ ~k,λ energy of the photon vacuum represents an absolute infinite energy scale with respect to which the photon states have energies that are higher by multiples of the photon energies ~ω , j =1, 2, . In this respect, two scenarios can be distinguished. { j ···} E does not depend on external parameters such as the dimensions of the box. This • 0 would for instance apply to a free electromagnetic field without any restrictions. From a quantum mechanical perspective only transitions between energy levels and energy differences are measurable. In that case merely the energy differences between the individual multi-photon energy eigenstates are relevant and the zero-point energy can simply be subtracted by defining that Hˆ 0 0 0 . This can be accomplished | i ≡ | i by normal ordering of the operator Hˆ (see 1.2.2): §

aˆ† aˆ~ N(ˆa† aˆ~ ) aˆ† aˆ~ , ~k,λ k,λ → ~k,λ k,λ ≡ ~k,λ k,λ

aˆ aˆ† N(ˆa aˆ† ) aˆ† aˆ , ~k,λ ~k,λ → ~k,λ ~k,λ ≡ ~k,λ ~k,λ which brings Hˆ in additive form:

2 ˆ ~ N(H) = ωk nˆ~k,λ . X~k λX=1

132 E does depend on external parameters such as the edge size L of the cubic box. A • 0 variation of these parameters can result in physically measurable energy shifts and forces (see the example below). In such cases we are at best allowed to subtract a universal (parameter-independent) energy, such as the zero-point energy belonging to an electromagnetic ﬁeld without any restrictions.

Casimir effect (Hendrik Casimir, 1948): consider two parallel, perfectly conducting perfectly conducting plates plates. These plates are positioned at a free field very short distance a of each other, such free field that the plates can effectively be considered as infinitely large on (a) length scales. O Since the zero-point energy between the plates differs from the one on the outside, a net attractive force between the plates is generated (see exercise 30). This is referred to as the Casimir effect. Experimentally it proved very difficult to align two plates with uniform micrometre precision. Even- tually it took 50 years to successfully verify ρ (free) E0 a the Casimir effect experimentally. This was ρE0 (free) ρ (a) done by switching from a two-plate config- E0 uration to a sphere-and-plate one: S.K. Lamoreaux covered in this way the a > 0.6 µm range in 1997, and U. Mohideen and A. Roy covered the a > 0.1 µm range in 1998. The Casimir effect is actually one of those rare purely quantum mechanical phenomena that extend beyond the molecular range.

By the way: the concept of zero-point energy might actually contain more profound physics than we anticipate at this moment. Not only energy diﬀerences have physical relevance, but also the absolute value of the energy. For instance, zero-point energy can have cosmological implications in view of the relation between energy and mass in general relativity. It is very well possible that the construction of a consistent fundamental theory of Nature requires the exact cancellation of the inﬁnities caused by the zero-point energies that originate from the various fundamental particles that are present in the theory.

4.2.4 Continuum limit

For completeness we list now the changes that would result from taking the size of the enclosure to inﬁnity (continuum limit). For our cube this continuum limit would imply

133 taking the limit L (see App. A.2). The ~k-sum extends over all integer numbers → ∞ νx,y,z =(L/2π)kx,y,z , which causes the ~k-spectrum to become very closely spaced (i.e. effectively continuous) for L and changes the sum into an integral: →∞ actually L V ======→∞ d~k . (355) −−−−→ (2π)3 X~k X~ν Z In addition, for L the commutation relations for creation and annihilation operators →∞ should satisfy the expressions for a continuous representation:

L (2π)3 ~ ~ ′ ~ ~ ˆ →∞ ~ aˆλ(k ), ˆaλ† ′ (k′) = δλ,λ δ(k k′) 1 aˆ~k,λ aˆλ(k ) . (356) − ⇒ −−−−→ r V ˆ The ﬁeld operator A~(~r, t) in equation (344) thus changes into

2 ˆ ~ ~ ~ ~ ~ † ~ ~ A(~r, t) = dk 3 ~ǫλ(~ek) aˆλ(k ) exp(ik ~r iωkt)+ˆaλ(k ) exp(iωkt ik ~r ) . 16π ǫ0 ωk · − − · λ=1 Z r h i X (357)

4.3 A new perspective on particle –wave duality ˆ The field operator A~(~r, t) contains the wave aspects that belong to the photon states. This reflects particle–wave duality in the sense that each creation and annihilation operator corresponds to a specific term in the plane-wave expansion of the vector potential A~(~r, t).

Let’s see how this diﬀers from the approach that we adopted in the previous chapter.

The positive-frequency terms: when attempting to set up a 1-particle QM we would • not have been dealing with the operatora ˆ exp(i~k ~r iω t), but rather with the ~k,λ · − k plane wave exp(i~k ~r iω t) as the corresponding stationary solution to the rel- ∝ · − k ativistic wave equation. By acting with the operators Eˆ = i~ ∂/∂t and ~pˆ = i~ ~ − ▽ on this plane wave we would have reached the conclusion that we were dealing with

a state with energy ~ωk and momentum ~~k. However, to the quantized electromagnetic theory a many-particle interpretation should be assigned, with the annihilation ~ operatora ˆ~k,λ exp(ik ~r iωkt) acting on multi-photon states. Applied to a multi- · − ~ photon state with n~k,λ photons in the 1-photon state with quantum numbers k and λ, this operator gives rise to a similar type of plane wave:

aˆ exp(i~k ~r iω t) n exp(i~k ~r iω t) , ~k,λ · − k → ~k,λ · − k where the size of the proportionality factorp depends on the occupation number of the given 1-photon state. In this context particle –wave duality refers to the fact

that the annihilation of a photon with positive energy ~ωk and momentum ~~k will always be accompanied by a plane-wave factor exp(i~k ~r iω t). ∝ · − k 134 The negative-frequency terms: the main diﬀerence between the 1-particle and many- • particle theory follows from the operatora ˆ† exp(iω t i~k ~r ). In the 1-particle ~k,λ k − · theory this would have corresponded to a plane wave exp(iω t i~k ~r ), which ∝ k − · we would have interpreted as a state with negative energy ~ω and momentum − k ~~k in the previous chapter. In the many-particle theory, however, the signs of − these quantum numbers are eﬀectively reversed, since we are dealing with photon creation rather than annihilation! The operatora ˆ† exp(iω t i~k ~r ) cor- ~k,λ k − · responds to the creation of a photon with positive energy ~ωk and momentum ~~k, which will always be accompanied by a plane-wave factor exp(iω t i~k ~r ). ∝ k − ·

Second quantization: the prescription for quantizing the electromagnetic field that we have just discussed is an explicit example of second quantization, as described in 1.5 for § the Schrödinger field. Starting from a classical wave equation we switched directly to a corresponding many-particle theory formulated in the Heisenberg picture. The classical vector potential A~(~r, t) was a classical, three-component function defined in each spacetime point. ˆ The quantized vector potential A~(~r, t) is a field operator that acts on the state functions in the Fock space for photons. In this field operator ~r and t are not quantum mechanical variables. They are simply parameters that parametrize the plane waves. In this approach ~r and t are treated on equal footing, in contrast to the non-relativistic QM where t is a parameter and ~r an eigenvalue belonging to the observable ~rˆ. This aspect is precisely what is needed for constructing the relativistic QM (see Ch. 5).

4.4 Interactions with non-relativistic quantum systems

In classical electrodynamics the influence of a classical electromagnetic field on a classical particle with mass m and charge q follows from minimal substitution. In the Coulomb gauge this implies that the momentum ~p of the classical particle should be replaced by ~p qA~(~r, t), where ~r is the coordinate of the particle. After this substitution ~p represents − the canonical momentum of the particle in the presence of the electromagnetic field.

The quantum mechanical Hamilton operator of the complete system is given by

Hˆ = Hˆ0 + Hê.m. + Vˆ (t) , with Vˆ (t) = Vînt(t)+ Vˆspin(t) , (358) where Hê.m. is the Hamilton-operator (346) of the free electromagnetic field inside an enclosure with volume V and periodic boundary conditions. The term Hˆ0 represents the Hamilton operator of the unperturbed non-relativistic particle, which can be a free particle or a particle that is bound by a potential. The interactions between the particle and the

135 electromagnetic ﬁeld are given by

q ˆ q2 ˆ Vˆ (t) = A~(~r,ˆ t) ~pˆ + A~ 2(~r,ˆ t) , (359) int − m · 2m ˆ ˆ ˆ ˆ e ~ S~ Vˆ (t) = ~ ~ (~r,ˆ t) , ~ = | | g . (360) spin − MS · B MS − 2m ~ The ﬁrst interaction is obtained by applying minimal substitution to the kinetic term ~pˆ 2/(2m) and by subsequently inserting the Coulomb condition ~ A~(~r, t) = 0. The sec- ▽ · ond interaction is the extra interaction between the magnetic ﬁeld and the spin-induced intrinsic magnetic dipole moment of the particle that we have encountered in chapter 3.

The interaction Vˆ (t) acts both on the charged-particle states, through ~rˆ and ~pˆ, and on ˆ the multi-photon states, througha ˆ~ anda ˆ† . The field operator A~(~r,ˆ t) provides the k,λ ~k,λ contact between the charged particle and the radiation field, with ~r being the position at which the particle experiences the interaction and with the interaction corresponding to the creation (emission) and annihilation (absorption) of photons. To phrase it differently: ˆ the field operator A~(~r, t) is the universal force carrier (mediator) of the electromagnetic interactions. Stronger radiation fields will result in stronger emission and absorption effects in view of the annihilation and creation factors nj and nj +1 in equation (351) being larger in that case. The difference between thesep two factorps will play an important role in the quantum mechanical phenomenon of spontaneous photon emission.

Vînt(t) in first-order perturbation theory: absorption/emission of one photon. In order to assess the implications of the quantization of the electromagnetic field, we consider electromagnetic processes that involve the absorption/emission of precisely one photon with quantum numbers ~k and λ. To further simplify the problem we assume for

convenience that the interaction Vˆint(t) can be treated as a weak perturbation and that

the magnetic interaction Vˆspin(t) can be neglected with repsect to Vînt(t) (as is true for the dipole approximation in 4.4.2). As a result, the spin of the particle can be effectively left § out in the analysis. Owing to the periodic time dependence of the electromagnetic field, we can apply periodic perturbation theory (see the lecture course Quantum Mechanics 2). ˆ At first order in perturbation theory, only the linear term in A~ contributes to 1-photon processes and effectively

Vˆ (t) Cˆ† exp( iω t) + Cˆ~ exp(iω t) , → ~k,λ − k k,λ k X~k λX=1 h i

(344),(359) q ~ with Cˆ~ ======aˆ† exp( i~k ~rˆ)~ǫ (~e ) ~p.ˆ (361) k,λ m 2V ǫ ω ~k,λ λ k − r 0 k − · ·

136 After all, at ﬁrst order in perturbation theory the number of photons changes by 1 due ˆ ± to the linear term in A~ and by 0, 2 due to the quadratic term.12 Take the system prior ± to the interaction to be in the following unperturbed initial state:

φ(0) = ψ , n , , (362) | i i | Ai|··· ~k,λ ···i ψ = unperturbed initial state of the charged particle with energy E , | Ai A , n , = photon state with n photons of the type (~k,λ) . |··· ~k,λ ···i ~k,λ Subsequently we wait a suﬃciently long time and ask ourselves the question what the transition probability will be to the ﬁnal state

φ(0) = ψ , n 1, , (363) | f i | Bi|··· ~k,λ ± ···i ψ = unperturbed ﬁnal state of the charged particle with energy E , | Bi B , n 1, = the same photon state with one (~k,λ)-photon more or less . |··· ~k,λ ± ···i The relevant matrix elements that determine this transition probability are

(361) q ~ (0) (0) Absorption : (C† ) ==== φ aˆ exp(i~k ~rˆ)~ǫ (~e ) ~pˆ φ ~k,λ fi m 2V ǫ ω f ~k,λ λ k i − r 0 k h | · · | i ~n (351) q ~k,λ ˆ ˆ ==== ψB exp(i~k ~r )~p ψA ~ǫλ(~ek) (364) − m s2V ǫ0 ωk h | · | i· if the photonic ﬁnal state is given by , n 1, , and |··· ~k,λ − ···i

(361) q ~ (0) (0) Emission : (C~ ) ==== φ aˆ† exp( i~k ~rˆ)~ǫ (~e ) ~pˆ φ k,λ fi m 2V ǫ ω f ~k,λ λ k i − r 0 k h | − · · | i ~(n +1) (351) q ~k,λ ˆ ˆ ==== ψB exp( i~k ~r )~p ψA ~ǫλ(~ek) (365) − m s 2V ǫ0 ωk h | − · | i· if the photonic ﬁnal state is given by , n +1, . By employing Fermi’s Golden |··· ~k,λ ···i Rule for periodic perturbation theory (see the lecture course Quantum Mechanics 2) this gives rise to the following transition rates (i.e. probabilities per unit of time) per photon state:

abs 2π 2 em 2π 2 w = (C† ) δ(E E ~ω ) , w = (C ) δ(E E + ~ω ) . ~k,λ ~ | ~k,λ fi| B − A − k ~k,λ ~ | ~k,λ fi| B − A k (366)

Depending on the size of n~k,λ a few different scenarios occur. 12At first order in perturbation theory the quadratic term contributes to 2-photon processes and to photon scattering off atoms. Note, though, that second-order perturbative corrections due to the linear term contribute to these processes as well.

137 If n is large enough that we may take n +1 n , then we are dealing • ~k,λ ~k,λ ≈ ~k,λ with a semi-classical scenario. In that case theq absorption/emissioq n of a single photon has a negligible impact on the radiation field, which effectively can be regarded as a classical field with a classical field strength A (ω ) 2~n /(V ǫ ω ). | 0 k | ≡ ~k,λ 0 k q For decreasing n the scenario becomes more and more quantum mechanical. • ~k,λ Absorption will still be approximated well by the semi-classical approach, since A (ω ) 2 n . However, this time the absorption of a photon does have an | 0 k | ∝ ~k,λ impact on the strength of the remaining radiation field. On the other hand, emission

will not be predicted adequately by the semi-classical approach, since n~k,λ +1 will

differ noticeably from n~k,λ . The most extreme differences with the semi-classical approach occur in the absence • of a radiation field , i.e. for n~k,λ = 0. In that case no transition is predicted to occur classically, whereas quantum mechanically there actually is the possibility of photon emission from an excited state! This purely quantum mechanical phenomenon is called spontaneous emission.

We observe that both types of emission are treated simultaneously in the quantum mechanical approach: spontaneous emssion if the right frequency component happens to be

absent in the radiation ﬁeld (n~k,λ = 0) and stimulated emission if a radiation ﬁeld is sup-

plied that does contain the right frequency component (n~k,λ > 0). In the latter case it follows directly from equation (365) that the transition probability increases for increasing n~k,λ . Hence the name stimulated emission.

4.4.1 Classical limit

Consider a monochromatic radiation field n . The classical electromagnetic description | ~k,λi applies if the quantization effects due to the commutation relations (342) can be negelected. This occurs when adding/removing a photon has no impact on the field strength, i.e. when ~ n~k,λ is large. Phrased more precisely: the number of (k,λ) -photons per characteristic volume ~3/ ~p 3 = 1/k3 should be very large. When acting on such a strong radiation | γ| field, the creation and annihilation operatorsa ˆ† anda ˆ~ can be represented by complex ~k,λ k,λ iδ iδ numbers e− k n respectively e k n (see 1.6.4), so that ~k,λ ~k,λ § p ~ p n~k,λ ~ˆ iδk ~ iδk ~ A(~r, t) n~k,λ ~ǫλ(~ek) e exp(ik ~r iωkt) + e− exp(iωkt ik ~r ) n~k,λ | i ≈ s2V ǫ0 ωk · − − · | i h i 2~n ~ ~k,λ A0(ωk) ~ǫλ(~ek) cos(k ~r ωkt + δk ) n~k,λ , A0(ωk) . ≈ | | · − | i | | ≡ s V ǫ0 ωk The action of a radiation field n on a charge carrier is in that case equivalent with | ~k,λi the classical action of a monochromatic plane wave.

138 We thus observe a clear distinction between the classical theories of radiation and particles (such as electrons).

The classical limit of the Schr¨odinger wave mechanics for an electron describes the • behaviour of that particle as governed by Newtonian mechanics, whereas a classical gas of such particles is obtained at low densities.

The classical limit of the quantum theory of radiation describes the behaviour of • very many photons at large densities by means of a classical electromagnetic wave that satisﬁes the Maxwell equations.

This also explains why ﬁrst the wave aspects of light and the particle aspects of electrons were observed, causing both phenomena to appear fundamentally diﬀerent.

4.4.2 Spontaneous photon emission (no exam material)

Using the density of states for photons from 4.2.2, the transition rate for spontaneous § emission of a photon with polarization λ and ~ek within a solid angle dΩ around ~eΩ is given by

spont.em. 2π ~ ~ 2 ~ WA B +γ ,dΩ = dΩ d( ωk) ρλ( ωk, Ω) (C~k,λ )fi δ(EB EA + ωk) → λ ~ | | − Z 2 (365),(352) q ω 2 AB ˆ ˆ . (367) ======2 3 2 ψB exp( iωAB ~eΩ ~r/c)~p ψA ~ǫλ(~eΩ) dΩ , 8π ~c m ǫ0 h | − · | i·

with the usual deﬁnition ω (E E )/~ AB ≡ A − B

Application 1: can a free particle with mass m = 0 and well-deﬁned momentum ~p ~~k 6 A ≡ A spontaneously emit a photon? Answer: no! Proof: the unperturbed spatial wave function of the particle prior to the interaction is

3/2 ψ (~r ) = (2π)− exp(i~r ~k ) . A · A As unperturbed ﬁnal state of the particle we consider an arbitrary eigenfunction of the momentum operator belonging to the momentum eigenvalue ~p ~~k : B ≡ B 3/2 ψ (~r ) = (2π)− exp(i~r ~k ) . B · B The relevant matrix element for the transition is then

3 ψ exp( i~k ~rˆ)~pˆ ψ = ~~k (2π)− d~r exp i~r [~k ~k ~k ] = ~~k δ(~k ~k ~k ) . h B| − · | Ai A · A − B − A A − B − Z

139 Based on equation (367) this implies that, for the transition to be possible, two conditions should be satisﬁed simultaneously . . . conservation of energy as well as momentum:

E = ~p 2 c2 + m2c4 = E ~ω = ~p 2 c2 + m2c4 ~ck and ~p = ~p ~~k . B B A − k A − B A − q q Both conditions can be merged to arrive at

momentum ~p 2 + m2c2 ======(~p ~~k)2 + m2c2 B A − energy ==== ~p 2 + m2c2 + ~2k2 2~k ~p 2 + m2c2 A − A q k =0 ~k ===6 ~p 2 + m2c2 = ~p ~p , ⇒ A A · k ≤ | A| q which is impossible if m = 0. This energy–momentum conservation argument holds just 6 as well for photon absorption and for 1-photon processes caused by Vˆspin(t).

Application 2: spontaneous transitions of 1-electron atoms in the dippole approximation. For spontaneous-emission transitions of relatively light 1-electron atoms (such as hydrogen) the wavelength of the emitted photon is much larger than the typical size of the emitting atom. For instance, the transition from the n = 2 to the n = 1 state in- 2 2 volves an energy difference ∆E = 3mec (Zα) /8. This corresponds to a wavelength 2 7 λγ = 2π/k = 2π~c/∆E (1.2/Z ) 10− m, which is much larger than the typical 10 ≈ × = (10− m) size of the charge distribution of the 1-electron atom, provided atoom O that Z is not too large. In the electric dipole approximation we exploit this fact by replacing the Fourier mode exp( i~k ~rˆ) by 1,ˆ since kr 1 across the full extent of the atom, − · ≪ and by leaving out the magnetic interaction. The justification for the latter approximation lies in the fact that the factor ~ǫ (~e ) ~pˆ in Vˆ (t) gives rise to (~/) effects, λ k · int O atoom whereas the factor ~~k ~ǫ (~e ) in Vˆ (t) leads to much weaker (~k) effects. × λ k spin O By means of the commutation relation

ˆ 2 ˆ ˆ ~p ˆ ˆ ∂H0 ~p Hˆ0 = + V (~r ) Hˆ0, ~r = i~ = i~ 2me ⇒ − ∂ ~pˆ − me the relevant matrix element for spontaneous photon emission in the dipole approximation is given by im ψ ~pˆ ψ = e ψ Hˆ , ~rˆ ψ = iω m ψ ~rˆ ψ iω m ~r . h B| | Ai ~ h B| 0 | Ai − AB e h B| | Ai ≡ − AB e BA From equation (367) it follows that in dipole approximation

e2ω3 W spont.em. AB ~r ~ǫ (~e ) 2 dΩ . (368) A B +γλ,dΩ 2 3 BA λ Ω → ≈ 8π ~c ǫ0 | · |

140 Subsequently we sum over all photon polarizations, using the completeness relation

2 3 2 ~ǫλ(~eΩ) IR (334) ǫ (~e )ǫ∗ (~e ) ======∈ ǫ (~e )ǫ (~e ) ==== δ e e , λ,j Ω λ,l Ω λ,j Ω λ,l Ω jl − Ω,j Ω,l λX=1 λX=1 and integrate over all directions ~eΩ while deﬁning the z-axis along ~rBA :

2π 1 e2ω3 e2ω3 ~r 2 spont.em. AB 2 2 AB | BA | 2 WA B +γ 2 3 dΩ ~rBA ~rBA ~eΩ = 2 3 dφ dcos θ sin θ → ≈ 8π ~c ǫ0 | | −| · | 8π ~c ǫ0 Z Z0 Z1 −

2 3 3 e ωAB 2 4 ωAB 2 spont.em. = 3 ~rBA = αem 2 ~rBA WA B +γ . (369) 3π~c ǫ0 | | 3 c | | ≈ →

In this expression αem is the electromagnetic ﬁne-structure constant, which is deﬁned in the usual way as e2 αem 1/137 . (370) ≡ 4πǫ0~c ≈

The average lifetime τA and decay width ΓA of the state A can be derived from

1 1 τA = = spont.em. , (371) ΓA WA Bj +γ j → P

which is summed over all allowed final states Bj for electric dipole transitions. These allowed final states have to satisfy the so-called dipole selection rules ∆ℓ = 1 and | | ∆m = 0 or 1, which is related to the spin of the emitted photon. Sometimes ~r = ~0 | ℓ| BA for all states with EB < EA . For instance this is the case for the 2s state of the hydrogen atom, for which no dipole transition to the 1s ground state is possible as both states have ℓ = 0. To determine the (longer) lifetime of such states one would have to take along the next term in the multipole expansion. This implies that the magnetic interaction cannot be discarded (consequence: magnetic dipole transitions) and that in Vˆ (t) the Fourier mode exp( i~k ~rˆ) should be replaced by 1ˆ i~k ~rˆ (consequence: int − · − · electric quadrupole transitions). For the 2s state of the hydrogen atom even these transitions are not possible: the 1-photon transition 2s 1s + γ is forbidden by angular → momentum conservation. The first allowed transition is the much weaker 2-photon transition 2s 1s + γ + γ , which explains why the lifetime of the 2s state (122 ms) is so much → larger than the one for the 2p state (1.6ns).

4.4.3 The photon gas: quantum statistics for photons

As ﬁnal example of the interaction between quantum systems and quantized electromagnetic ﬁelds we consider the quantum statistics for photons. From equations (364)–(366) a

141 condition can be derived for thermal equilibrium between an atomic heat bath at temperature T and electromagnetic radiation with quantum numbers ~k and λ (see exercise 31). From this equilibrium condition it follows that the average occupation number of the corresponding 1-photon state at temperature T =1/(kBβ) is given by 1 n¯ = , (372) ~k,λ exp(β~ω ) 1 k − which we recognize as a Bose–Einstein distribution with α = 0. This is exactly the same distribution as derived in exercise 14 for a canonical ensemble of systems that each consist of a single linear harmonic oscillator. The fact that photons in general do not allow for a grand canonical treatment has to do with the absence of a photon conservation law in most quantum mechanical settings.13 As a result of absorption by the atoms in the heat bath, the photons do not migrate to the heat bath as particles and exclusively contribute to the thermodynamic energy balance. The entropy of the heat bath is in fact independent of the number of photons. The absence of a photon conservation law also implies that no Bose–Einstein condensation of low-energy photons can occur in such situations. It will be entropically favourable for the atomic heat bath to simply absorb such photons: the condensate only involves a single quantum state whereas the number of heat-bath states increases upon energy absorption.

Let’s now enclose the electromagnetic ﬁeld inside a black body with walls that can absorb/emit photons of all energies. In this idealized situation we are dealing with an ideal photon gas. The energy distribution of the radiation ﬁeld per unit of energy and volume then reads 2 3 (372) 1 E (352) 8π E U(E) ==== dΩ ρ (E, Ω) ==== . (373) V exp(βE) 1 λ h3c3 exp(βE) 1 − Z λX=1 − From this expression we can derive Planck’s law for the energy distribution per unit of frequency ν = E/h and volume:

3 dE(ν) (373) 8πhν 1 U(ν) = U(E) ==== , (374) dν c3 exp(hν/k T ) 1 B − as well as the law of Stefan–Boltzmann for the total energy density:

∞ 4 4 ∞ 3 2 4 (373),x=βE k T x π k U = dE U(E) ======B dx = B T 4 . (375) tot π2~3c3 exp(x) 1 15~3c3 Z0 Z0 −

13We used to think that a photon gas with a conservation law for the total number of photons was not realizable in the lab. However, in 2010 the situation changed. The photon gas inside the microresonator described in exercise 21 eﬀectively satisﬁes a photon conservation law and as such allows the formation of a Bose–Einstein condensate.

142 Experimental realization: use a large cavity (furnace) with a tiny hole in it. If the walls of the cavity are sufficiently rough and opaque, any light entering the cavity will be reflected very many times before it will be able to escape the cavity through the tiny hole. The light then has a high probability of being absorbed by the walls before it manages to escape. In good approximation the tiny hole acts as a black body. By varying the temperature of the cavity, Planck’s law of black-body radiation can be verified experimentally.

Another almost perfect example of Planck’s spectrum for black-body radiation is provided by the present-day cosmic microwave background (CMB), which corresponds to a temperature of T =2.725 K. The present-day CMB radiation has evolved from a snapshot of our universe roughly 380,000 years after the Big Bang, when the universe became transparent to radiation. Due to the expansion of the universe, the earlier (distant) universe was of course warmer than the universe today. However, the corresponding radiation is redshifted due to that same expansion, causing it to appear as having the same temperature as the universe today!

143 5 Many-particle interpretation of the relativistic QM

Motivated by the quantization of the electromagnetic ﬁeld, this chapter will be devoted to assigning a many-particle interpretation to the solutions of the relativistic wave equations that were constructed in chapter 3. More extended versions of the material covered in this chapter can be found in Schwabl (Ch.13) and Merzbacher (Ch.24).

Negative-energy states and the 1-particle relativistic QM: in chapter 3 the at- tempts to construct a relativistic 1-particle quantum theory always resulted in a set of hard to interpret negative-energy states. In the bosonic case, such as in the Klein–Gordon theory, this resulted in unsurmountable problems that basically disqualified the theory as a viable 1-particle quantum theory. In the fermionic case, such as in the free Dirac theory, Pauli’s exclusion principle provided a way out by assuming that all 1-fermion negative- energy states are occupied in the vacuum state of the universe. These states in the Dirac sea behaved dynamically as if we were dealing with objects with opposite charge. This can be highlighted by looking at the relativistic 1-electron atom, which has bound states with positive energy. If we would invert the nuclear charge (Z Z), only bound states → − with negative energy occur. This is consistent with the picture that the positively charged nucleus effectively repels the negative-energy electrons and that upon inverting the nuclear charge the negative-energy electrons are effectively attracted. Another way to read off this opposite charge is by selecting negative-energy states in the Dirac equation (308) by means of the substitution E = E mc2. In that case we obtain the same eigenvalue equation NR − as in 3.2.5 (I) with ψ ψ , E E , ~pˆ ~pˆ and q q. It was shown in § U ↔ D NR → − NR → − → − exercise 27 how a negative-energy state can indeed be turned into a positive-energy state with opposite charge by performing charge conjugation.

Hidden inside the free Dirac theory apparently resides the description of a second type of particle with opposite charge. By employing hole theory (see the discussion in 1.7.1), § these particles with opposite charge could be made manifest as holes in the Dirac sea. However, hole theory will not be able to describe all observed phenomena involving the oppositely charged particles.

A conﬂict with hole theory: the occurrence of the β+ decay process p n+e+ +ν in → e the decay of proton-rich nuclei actually contradicts this hole-theory interpretation. Within hole theory we would have liked to interpret the produced e+ as a hole in the Dirac sea. However, the creation of a hole in the Dirac sea should automatically be accompanied by the simultaneous creation of a positive-energy electron (see the discussion on electron–hole excitations in 1.7.1), which evidently is not the case in the observed β+ decay process! §

144 Many-particle interpretation of the relativistic QM: all this seems to suggest that it may be better to not set up the relativistic QM as a 1-particle theory, but rather as a many-particle theory in analogy to the treatment of the electromagnetic ﬁeld. This idea does not contradict with the non-relativistic QM. After all, as noted earlier, photons are massless and should therefore always be treated relativistically. These photons can (in most situations) be created and annihilated without energy threshold. Massive particles, however, do have an energy threshold as a direct consequence of their non-zero rest mass:

for kinetic energies mc2 such systems behave non-relativistically, i.e. they corre- • ≪ spond to a ﬁxed number of particles;

for kinetic energies = (mc2) the many-particle aspects manifest themselves. • O

The many-particle theory will be set up along the lines worked out for the electromagnetic field: the field equation will be solved by performing a Fourier decomposition (plane- wave expansion) and the Hamiltonian belonging to the field equation will be expressed as an infinite set of independent linear harmonic oscillators. Finally, this Hamiltonian will be quantized in the Heisenberg picture and the fundamental energy quanta will be interpreted as particles. Particle–wave duality then follows automatically from the fact that the creation/annihilation of a particle is always accompanied by a corresponding plane-wave factor, as can be read off from equation (344) in the electromagnetic case.

From this point onward the 1-particle quantum mechanical approach makes way for a many-particle formulation based on second quantization. This many-particle formulation is referred to as relativistic quantum ﬁeld theory and is usually described in terms of path integrals or Lagrangian densities for continuous systems. In this lecture series we will content ourselves with deriving the salient aspects of quantization while avoiding ﬁeld- theoretic bells and whistles as much as possible.

5.1 Quantization of the Klein–Gordon ﬁeld

Again we start oﬀ by considering the Klein–Gordon equation for free spin-0 particles with rest mass m and charge q. For the derivation of the many-particle formulation we ignore the problems with the 1-particle quantum mechanical interpretation and treat the Klein– Gordon equation as a relativistic classical wave equation.

The classical energy density: in order to highlight the analogy with the electromagnetic case we will rescale the Klein–Gordon field according to ψ(x) = √2mc2 φ(x)/~. This rescaled Klein–Gordon field φ(x) satisfies the field equation

m2c2 1 ∂2 2 m2c2 + φ(x) = ~ + φ(x)=0 (376) ~2 c2 ∂t2 − ▽ ~2

145 and has a corresponding classical energy density14 (without proof)

2 4 ∂ ∂ 2 ~ ~ m c ρ (x) = φ∗(x) φ(x) + c φ∗(x) φ(x) + φ∗(x)φ(x) . (377) H ∂t ∂t ▽ · ▽ ~2 In this form the similarity with the classical electromagnetic energy density is apparent. The first two terms are the analogues of the electric and magnetic contributions to the energy density. The last term is a rest-mass term, which is absent for the massless photons. The other differences between both energy densities reflect that photons have spin 1, whereas the particles belonging to the Klein–Gordon theory have spin 0 (see 3.1). § Remark: in the relativistic quantum-field-theory approach all this is handled more systematically by working with Lagrangian densities for continuous systems, using φ(x) and

φ∗(x) as generalized coordinates. Everything else follows from this Lagrangian density. The Klein–Gordon equation simply follows from the Euler–Lagrange equation (equation of motion) that belongs to this Lagrangian density. Moreover, if we go from the Lagrangian density to the corresponding Hamiltonian density of the system we obtain an expression that coincides with the energy density given above.

Plane-wave expansion and quantization: in analogy to the electromagnetic case we also solve the Klein–Gordon equation inside a ﬁnite cubic enclosure with periodic boundary conditions: ~ 1 φ(x) = u~p (~x ) a~p (t)+ c~p (t) , u~p (~x ) = exp(i~p ~x/~) . (378) 2E~p √V · X~p h i p The positive- and negative-frequency Fourier coeﬃcients a~p (t) and c~p (t) satisfy

a (t) = a exp( iE t/~) and c (t) = c exp(iE t/~) , ~p ~p − ~p ~p ~p ~p 2π~ with ~p = ~ν , E m2c4 + ~p 2c2 > 0 (ν = 0, 1, 2, ) . (379) L ~p ≡ x,y,z ± ± ··· p The Fourier components u~p (~x ) are normalized in the usual way:

(E.4) ′ d~xu~p (~x )u∗~p ′ (~x ) ==== δ~p, ~p . (380) VZ The total energy inside the volume V then simply reads

(377),(380) E (t) = d~xρ (x) ======~p a (t) c (t) 2 + a (t)+ c (t) 2 H H 2 | ~p − ~p | | ~p ~p | VZ X~p = E a (t) 2 + c (t) 2 = E a 2 + c 2 = . (381) ~p | ~p | | ~p | ~p | ~p | | ~p | H X~p X~p 14 For φ(x) IR this energy density should be multiplied by a factor 1/2, since the number of degrees ∈ of freedom is halved in that case.

146 Next this superposition of vibrational modes is quantized in the Heisenberg picture in analogy with equations (337)–(341) for the electromagnetic ﬁeld:

a (t) aˆ (t) aˆ exp( iE t/~) , ~p → ~p ≡ ~p − ~p ˆ ˆ ~ c~p (t) b† ~p (t) b† ~p exp(iE~p t/ ) . (382) → − ≡ − In order to be able to give a correct particle interpretation to the negative-energy solutions, it is essential that the second operator is a creation operator if the first one is an annihilation operator. The scalar Klein–Gordon field then becomes a scalar field operator

~ ˆ ˆ φ(x) = u~p (~x ) aˆ~p (t) + b† ~p (t) 2E − ~p ~p X ~ h i (383) p ~ ˆ ~ = aˆ~p exp( ip x/ ) + b†~p exp(ip x/ ) 2V E~p − · · X~p h i p with p0 E /c = m2c2 + ~p 2 , (384) ≡ ~p p satisfying the same wave equation as the classical Klein–Gordon ﬁeld.

Particle interpretation: the following particle interpretation can be assigned to the quantized Klein–Gordon theory (see the many-particle operators listed on the next page).

The operatorsa ˆ~p anda ˆ†~p annihilate and create spin-0 particles with momentum ~p and ˆ ˆ positive energy E~p . The operators b~p and b†~p annihilate and create the corresponding antiparticles with the same spin, momentum and positive energy, but with opposite charge. The field operator φˆ(x) acts on the state functions in the Fock space belonging to both the spin-0 particles and antiparticles.15 For a sensible treatment of negative energy solutions to the Klein–Gordon equation, particles and antiparticles are thus described in a joint way in the many-particle formulation of the relativistic QM. If a positive-energy solution corresponds to the annihilation/creation of a positive-energy particle, a negative-energy solution corresponds to the creation/annihilation of a positive-energy antiparticle with opposite charge. Particles and antiparticles can be regarded as identical particles with opposite values for the charge quantum number, whereas the field operator φˆ(x) corresponds to one specific value of this quantum number. This quantum number indicates how the particles interact with the outside world. We distinguish three types of (conserved) charge: electric charge, weak charge (weak isospin) and strong charge (colour charge), belonging to the three fundamental non-gravitational forces that are known to us.

15 If φ(x) IR, then ˆb† =a ˆ† . This lifts the distinction between particles and antiparticles, which ∈ ~p ~p therefore should carry zero charge.

147 Motivation for this particle interpretation: just like in the electromagnetic case, the creation and annihilation operators satisfy bosonic commutation relations:

ˆ ˆ ′ ˆ aˆ~p (t), aˆ†~p ′ (t) = b~p (t), b†~p ′ (t) = δ~p, ~p 1 , all other commutators vanish . (385) Now we introduce the number operators

(+) ( ) nˆ aˆ† (t)ˆa (t) =a ˆ† aˆ andn ˆ − ˆb† (t)ˆb (t) = ˆb† ˆb (386) ~p ≡ ~p ~p ~p ~p ~p ≡ ~p ~p ~p ~p

for counting particles and antiparticles, respectively. In analogy to equation (343), this allows us to write the total Hamilton operator of the free non-interacting many-particle system in a compact way as

ˆ ˆ ˆ (+) ( ) ˆ H = E~p aˆ†~p(t)â~p(t) + b ~p(t)b† ~p(t) = E~p nˆ~p +n ˆ~p− + 1 . (387) − − X~p X~p The energy spectrum is indeed nicely bounded from below. Again we obtain a zero-point vacuum energy that is infinite and positive. The rejected “probability density” of 3.1 is § given a completely different meaning in the many-particle formulation. It is related to the total charge operator of the many-particle system, since

iqc Qˆ d~x φˆ†(x)∂ φˆ(x) ∂ φˆ†(x)φˆ(x) tot ≡ ~ 0 − 0 VZ

(383), 1st q ˆ ˆ ˆ ˆ ==== aˆ†~p (t)+ b ~p (t) aˆ~p (t) b† ~p (t) + aˆ†~p (t) b ~p (t) aˆ~p (t)+ b† ~p (t) 2 − − − − − − ~p X (385) (+) ( ) ˆ (+) ( ) ˆ ˆ ==== q nˆ~p nˆ −~p 1 = q nˆ~p nˆ~p− 1 = Qtot (388) − − − − − X~p X~p indeed counts particles with charge q and antiparticles with charge q. The Lorentz − invariant continuity equation that was derived in 3.1 in fact does not represent conser- § vation of probability, but rather conservation of total charge of the many-particle system. The total momentum operator of the free non-interacting many-particle system is given by the additive expression (without proof)

~ˆ (+) ( ) P = ~p nˆ~p +n ˆ~p− . (389) X~p

The particle interpretation that was given on the previous page can now be read oﬀ directly from the three many-particle operators (387)–(389).

148 Locality: physical observables at different spatial positions should mutually commute and as such be measurable simultaneously. After all, the results of a repeated quantum mechanical measurement in a certain spatial point should not be affected by simultaneously performing a repeated quantum mechanical measurement in another spatial point, irrespective of how far the two points are apart. This is called locality or microcausality. If these measurements would influence each other, it would constitute a transfer of information that exceeds the speed of light, which is forbidden by the theory of relativity. During one measurement we would be able to observe the fact that the other measurement is taking place. In the coordinate representation the physical observables can be written in terms of the field operators (383). Hence, locality should follow directly from the commutation relations that pertain to these field operators. The relevant commutators are

(383),(385) φˆ(~x, t), φˆ(~x ′, t) = φˆ†(~x, t), φˆ†(~x ′, t) ======0 , (390)

~2 u~p (~x )u∗~p ′ (~x ′) (385) ˆ ˆ ˆ ˆ ′ φ(~x, t), φ†(~x ′, t) = aˆ~p (t)+ b† ~p (t) , aˆ†~p ′ (t)+ b ~p (t) ==== 0 . ′ − − ~p, ~p ′ 2 E~p E~p X p These commutators nicely vanish in all required situations. By the way, this would not have been the case if the creation and annihilation operators would have satisﬁed fermionic anticommutation relations.

5.2 Quantization of the Dirac ﬁeld

The classical energy density: consider a Dirac ﬁeld ψ(x) that satisﬁes the classical Dirac equation for free spin-1/2 particles with rest mass m and charge q:

∂ i~ ∂/ mc ψ(x)=0 i~ + i~c ~α ~ β mc2 ψ(x) =0 . (391) − ⇒ ∂t · ▽ − The corresponding classical energy density is given by

∂ ~ 2 ρ (x) = ψ†(x) i~ ψ(x) = ψ†(x) i~c~α + β mc ψ(x) . (392) H ∂t − · ▽ Plane-wave expansion and quantization: subsequently the Dirac equation is solved inside a ﬁnite cubic enclosure with periodic boundary conditions. This results in the following plane-wave decomposition:

2 mc + ψ(x) = u~p (~x ) uλ (~p )a~p,λ(t) + uλ−(~p )c~p,λ(t) (393) s E~p X~p Xλ h i ~ exp(i~p ~x/ ) + · u (~p )a~p,λ exp( iE~p t/~) + u−(~p )c~p,λ exp(iE~p t/~) , 2 λ λ ≡ V E~p /mc − X~p Xλ h i p 149 with ~p and E~p the same as for the Klein–Gordon ﬁeld [see equation (379)]. The Dirac

spinors uλ±(~p ) satisfy ~ˆ ˆ ~ S ~puλ±(~p ) = λ ~puλ±(~p ) and ( p/ mc)uλ±(~p ) =0 , (394) · ± − 0 where λ denotes the helicity eigenvalue (being +1/2 or 1/2), p = E~p /c the energy − ± ± eigenvalue and ~p = ~p the momentum eigenvalue. For free spin-1/2 particles it is derived ± ˆ in exercise 26 that H,ˆ ~pˆ and S~ ~pˆ indeed form a commuting set of operators. These Dirac · spinors are normalized according to the normalization condition in equation (300):

E~p u±(~p )† u±′ (~p ) = δ ′ and u±(~p )† u∓′ (~p )=0 . (395) λ λ mc2 λ,λ λ λ

By means of the normalization condition (380) for the Fourier components u~p (~x ), the total energy inside the volume V reads

(392),(393) 2 + (t) = d~xρH (x) ======mc uλ (~p )† a∗~p,λ(t) + uλ−(~p )† c~p,λ∗ (t) H ′ × VZ X~p Xλ,λ h i

+ (395) 2 2 u ′ (~p )a ′ (t) u−′ (~p )c ′ (t) ==== E a (t) c (t) , × λ ~p,λ − λ ~p,λ ~p | ~p,λ | − | ~p,λ | h i X~p Xλ which results in the additive expression

= E a 2 c 2 . (396) H ~p | ~p,λ| − | ~p,λ| X~p Xλ Looking at this minus sign, it is clear that it makes no sense to perform the quantization on the basis of bosonic creation and annihilation operators. The classical total energy is not bounded from below and can therefore not be quantized as in the Klein– Gordon theory. To compensate this minus sign we are forced to resort to the alternative many-particle formulation based on creation and annihilation operators that satisfy fermionic anticommutation relations.

In this case the quantization in the Heisenberg picture goes as follows:

a (t) aˆ (t) aˆ exp( iE t/~) , ~p,λ → ~p,λ ≡ ~p,λ − ~p ˆ ˆ ~ c~p,λ (t) b† ~p,λ (t) b† ~p,λ exp(iE~p t/ ) . (397) → − ≡ − The Dirac field becomes a field operator that satisfies the Dirac equation:

2 ˆ mc + ˆ ψ(x) = u~p (~x ) uλ (~p )ˆa~p,λ(t) + uλ−(~p )b† ~p,λ(t) E~p − λ s X~p X h i , (398) 2 mc + ~ ˆ ~ = uλ (~p )ˆa~p,λ exp( ip x/ )+ uλ−( ~p )b†~p,λ exp(ip x/ ) sV E~p − · − · X~p Xλ h i 150 with as before p0 E /c = m2c2 + ~p 2 . (399) ≡ ~p p

Particle interpretation and its motivation: the operatorsa ˆ~p,λ anda ˆ†~p,λ annihilate and create spin-1/2 particles with momentum ~p, positive energy E~p and helicity λ. The ˆ ˆ operators b~p,λ and b†~p,λ annihilate and create the corresponding antiparticles with the same spin, momentum, positive energy and helicity, but with opposite charge. As indicated, in order to have an energy spectrum that is bounded from below we require the creation and annihilation operators to satisfy fermionic anticommutation relations:

ˆ ˆ ′ ′ ˆ aˆ~p,λ(t), aˆ†~p ′,λ′ (t) = b~p,λ(t), b†~p ′,λ′ (t) = δ~p, ~p δλ,λ 1 , all other anticommutators vanish . (400) Again this implies an identical-particle approach in which particles and antiparticles are described in a joint way in the relativistic QM. Next we introduce the number operators

(+) ( ) nˆ aˆ† (t)ˆa (t)=a ˆ† aˆ andn ˆ − ˆb† (t)ˆb (t) = ˆb† ˆb (401) ~p,λ ≡ ~p,λ ~p,λ ~p,λ ~p,λ ~p,λ ≡ ~p,λ ~p,λ ~p,λ ~p,λ

to count the number of particles and antiparticles. The total Hamilton operator of the free non-interacting many-particle system can be written compactly as

ˆ ~ ∂ ˆ (380),(395),(398) ˆ ˆ d~x ψ†(x) i ψ(x) ======E~p aˆ†~p,λ(t)ˆa~p,λ(t) b ~p,λ(t)b† ~p,λ(t) ∂t − − − VZ X~p Xλ (+) ( ) ˆ (+) ( ) ˆ ˆ = E~p nˆ~p,λ +n ˆ −~p,λ 1 = E~p nˆ~p,λ +n ˆ~p,λ− 1 = H . (402) − − − X~p Xλ X~p Xλ The energy spectrum is indeed nicely bounded from below thanks to the anticommutation ˆ ˆ relations for b†~p,λ(t) and b~p,λ(t). In the many-particle formulation, the probability density of 3.2.1 receives the meaning of total charge operator of the many-particle system, since §

ˆ ˆ ˆ (380),(395),(398) ˆ ˆ Qtot q d~x ψ†(x)ψ(x) ======q aˆ†~p,λ(t)ˆa~p,λ(t) + b ~p,λ(t)b† ~p,λ(t) ≡ − − VZ X~p Xλ

(400),(401) (+) ( ) ======q nˆ nˆ − + 1ˆ = Qˆ (403) ~p,λ − ~p,λ tot X~p Xλ indeed counts particles with charge q and antiparticles with charge q. Finally, the − total momentum operator of the free non-interacting many-particle system is given by the additive expression ~ˆ (+) ( ) P = ~p nˆ~p,λ +n ˆ~p,λ− . (404) X~p Xλ

151 Vacuum energy: in the expression for the Hamilton operator given above the vacuum has an infinite negative energy, but it does define a lowest-energy state (ground state). As a result of the fermionic quantization conditions the vacuum has a zero-point energy per vibration mode that is opposite to the corresponding zero-point energy for bosonic quantization conditions. It is very well possible that nature has equal amounts of bosonic and fermionic degrees of freedom, allowing for an automatic cancellation of the infinities in the zero-point energies. This idea is for instance used in supersymmetry , which links bosonic to fermionic degrees of freedom by means of a symmetry. Quantum mechanically speaking this cancellation of infinities is not essential, but cosmologically it is since an energy density curves spacetime.

Locality: the requirement of locality forces us to impose the full set of anticommutation relations (and not just the one for the b-operators). In exercise 32 it will be shown that in that case

ˆ ˆ ′ ˆ ˆ ψi(~x, t), ψi (~x ′, t) = ψi†(~x, t), ψi†′ (~x ′, t) = 0 , (405) ψˆ (~x, t), ψˆ†′ (~x ′, t) = δ ′ δ(~x ~x ′) 1ˆ (i, i′ =1, , 4) , i i ii − ··· causing the relevant anticommutators of the Dirac fields to vanish in all required situations. This would not have been the case if the creation and annihilation operators would have satisfied bosonic commutation relations. In this context, an important thing to note is that physical observables should consist of even numbers of Dirac fields, since physical observations should not change under 2π rotations of space while Dirac spinors flip sign (see exercixe 25). Products of even numbers of Dirac fields give rise to commutators of observables that can be written in terms of anticommutators of Dirac fields (see the trick on p. 10), allowing locality to be realized by means of either anticommutation relations or commutation relations.

Spin – statistics theorem: spin and statistics are related in the relativistic QM. The quantization of relativistic wave equations belonging to arbitrary spins will result in the generic statement that bosons should have integer spin and fermions half integer spin.

In order to explicitly prove this theorem one should ﬁrst of all demand the energy spectrum to be bounded from below, This will determine whether the ˆ ˆ creation and annihilation operators b†~p,λ and b~p,λ satisfy commutation relations or anticommutation relations. From the requirement of locality then follows that

aˆ†~p,λ and aˆ~p,λ should satisfy the same operator algebra and that both sets of operators should also commute or anticommute mutually.

152 5.2.1 Extra: the 1-(quasi)particle approximation for electrons

Starting from the many-particle theory for electrons and positrons (anti-electrons) we will now try to understand the success of the 1-particle Dirac theory. Deﬁne to this end the

electron vacuum 0 , with ψˆ(x) 0 0 and 0 0 1 . (406) | ei | ei ≡ h e| ei ≡

This implies that aˆ~p,λ 0e = ˆb† 0e = 0, i.e. this vacuum contains no electrons. ~p,λ∀ | i ~p,λ | i However, all possible 1-positron states are occupied in this vacuum, causing it to contain an inﬁnite number of positrons. We are dealing here with a quasi-particle vacuum belonging to the quasi-particle annihilation operator ψˆ(x) in the Heisenberg picture. In view of the exclusion principle, this way of constructing the electron vacuum exclusively works for fermions and not for bosons.

Both adding an electron to the electron vacuum and removing a positron from the electron vacuum can be regarded as the creation of a kind of 1-electron state, where in the latter case we are dealing with a negative effective energy but electron-like effective charge e −| | with respect to the vacuum. In this quasi-particle picture an arbitrary 1-electron state can be defined as a

1-charge state 1 , with 1 d~x ψˆ†(x)ψ (x) 0 . (407) | ei | ei ≡ e | ei VZ

This 1-charge state is deﬁned in the Heisenberg picture, just as the ﬁeld operator ψˆ(x).

This automatically implies that the time dependence of ψˆ†(x) should be compensated

by the time dependence of the spinor coefficient ψe(x). The coefficient ψe(x) will be identified with the spinor wave function in the 1-electron Dirac theory. In exercise 32 it

will be proven that the states ψˆ†(x) 0 have the following orthonormality property: | ei

0 ψˆ (~x, t)ψˆ†′ (~x ′, t) 0 = δ ′ δ(~x ~x ′) 1ˆ (i, i′ =1, , 4) , (408) h e| i i | ei ii − ··· from which we can deduce that

1 1 = d~x d~x ′ ψ†(~x, t) 0 ψˆ(~x, t)ψˆ†(~x ′, t) 0 ψ (~x ′, t) = d~xψ†(~x, t)ψ (~x, t) , h e| ei e h e| | ei e e e VZ VZ VZ

(407) ψ (~x, t) = d~x ′ 0 ψˆ(~x, t)ψˆ†(~x ′, t) 0 ψ (~x ′, t) ==== 0 ψˆ(~x, t) 1 . (409) e h e| | ei e h e| | ei VZ

In this way the normalization of the spinor wave function ψe(x) follows automatically from the normalization of the 1-charge state 1 . | ei

153 The charges of the above-mentioned many-particle states are given by

(407) (410) Qˆ 1 ==== d~x Qˆ ψˆ†(x) 0 ψ (x) ==== d~x Qˆ , ψˆ†(x) 0 ψ (x) tot | ei tot | ei e tot | ei e Z Z V V

(403) (407) ==== e d~x ψˆ†(x)ψ (x) 0 ==== e 1 , (411) −| | e | ei −| || ei VZ

(+) ( ) using thatn ˆ 0 =0 0 andn ˆ − 0 = 0 , as well as the relations ~p,λ | ei | ei ~p,λ | ei | ei

(+) ( ) (400),(401) nˆ nˆ − + 1ˆ, aˆ† ′ ′ (t) ======a ˆ† ′ ′ (t) ~p,λ − ~p,λ ~p ,λ ~p ,λ ~p λ X X and (+) ( ) (400),(401) nˆ nˆ − + 1ˆ, ˆb ′ ′ (t) ======ˆb ′ ′ (t) . ~p,λ − ~p,λ ~p ,λ ~p ,λ ~p λ X X The terminology 1-charge state for 1 thus makes good sense just like the designation | ei “vacuum” for 0 , which ﬁnds further support in the expression for the total energy: | ei (402) (+) ( ) Hˆ 0 ==== E nˆ +n ˆ − 1ˆ 0 = 0 0 . (412) | ei ~p ~p,λ ~p,λ − | ei | ei X~p Xλ Finally we have

(409) i~c~α ~ + β mc2 ψ (x) ======0 i~c~α ~ + β mc2 ψˆ(x) 1 − · ▽ e h e| − · ▽ | ei Dirac eqn. ∂ (409) ∂ ======0 i~ ψˆ(x) 1 ==== i~ ψ (x) . (413) h e| ∂t | ei ∂t e

Hence, ψe(x) satisﬁes the Dirac equation, it is normalizable and it contains negative- energy components. As such, ψe(x) can be identiﬁed with the spinor wave function in the 1-electron Dirac theory.

The 1-particle Dirac theory is therefore related to a specific 1-(quasi)particle approximation of the many-particle theory. This approximation can be applied successfully as long as no fluctuations of order m c2 occur in the energy on time scales of order ∆t = (~/m c2) e O e or on length scales of order ∆r = (~/m c). If such fluctuations would occur, effects from O e the creation/annihilation of electrons and positrons should be taken into account. For instance, electron–positron annihilation can occur, i.e. the emission of a photon caused by the annihilation of an electron with a positron from the electron vacuum. The inverse of this photon-emission process is called pair creation. For calculating such effects

154 quantum field theory is the tool of choice, which describes all particles (electrons, positrons, photons, . . . ) by means of field operators and represents the field equations including interactions by means of Lagrangians. These interactions have their origin in so-called gauge symmetries and the associated charge-conservation laws, such as conservation of total electric charge in the case of electromagnetic interactions.

155 A Fourier series and Fourier integrals

Fourier decompositions are used quite frequently in these lecture notes. We brieﬂy list the relevant aspects of these decompositions in this appendix.

A.1 Fourier series

Systems with a ring-shaped structure and systems that are enclosed spatially can be described in terms of state functions with a certain degree of periodicity. Consider to this end a 1-dimensional function f(x), for which the function as well as the spatial derivative are stepwise continuous on the interval x L/2, L/2 and which additionally satisﬁes ∈ − the spatial periodicity condition f(x)= f(x+ L). From Fourier analysis it follows (see the lecture course “Trillingen en Golven”) that such a function can be decomposed in terms of a basis of periodic plane waves (Fourier modes): 1 1 fn(x) = exp(2πinx/L) exp(ikx) (n = 0, 1, 2, ) . (A.1) √L ≡ √L ± ± ··· We readily recognize these functions as being momentum eigenstates belonging to the quantized momentum eigenvalues ~k. This basis satisﬁes the orthonormality relation

x0 +L 1 dx exp ix[k k′] = δ ′ , (A.2) L − kk Z x0 where x can be chosen arbitrarily. Since x is usually restricted to the interval L/2, L/2 , 0 − a handy choice is x = L/2. By employing the completeness relation for periodic Fourier 0 − modes, the periodic function f(x) can be written as a Fourier series:

∞ A f(x) = n exp(2πinx/L) (A.3) √ n= L X−∞

A0 ∞ An + A n An A n = + − cos(2πnx/L) + i − − sin(2πnx/L) √ √ √ L n=1 L L X h i

B ∞ 2 0 + B cos(2πnx/L)+ C sin(2πnx/L) , (A.4) ≡ √ L n n L n=1 r X h i with L/2 1 An = dx f(x) exp( 2πinx/L) . (A.5) √L − L/Z 2 − Periodic systems vs enclosed systems: the close relation between periodic systems and systems that are contained inside an enclosure can be seen as follows. If we assume the im- penetrable walls of the enclosure to be positioned at x = 0 and x = L, the state functions

i have to satisfy the continuity condition ψ(0) = ψ(L) = 0 (see 2.5). Inside the enclo- § sure this function coincides with a periodic function f(x) = f(x +2L) with odd parity, so that f(x) = f( x). As a result of this extra condition, only the standing waves − − sin(πnx/L) for n =1, 2, contribute to the Fourier series (A.4), which indeed guar- ∝ ··· antees that f(0) = f(L) = 0. The periodicity length is 2L now and at the same time n is allowed to take on positive values only. The number of allowed quantized k values is in this case identical to the number of k values for a system with periodicity length L, with the exception of the absence of the quantum number k = 0. The presence of the edge of the box manifests itself in the absence of the constant Fourier mode with quantum number k =0. The inﬂuence of the edge of the enclosure is only felt when the quantum states with the lowest absolute values for the momentum (and thus the largest de Broglie wavelengths) contribute signiﬁcantly to the description of the system. This happens for instance in bosonic gases at low temperatures (see Ch. 2).

A.2 Fourier integrals

The Fourier decomposition of non-periodic quadratically integrable functions can be obtained by taking the continuum limit L of the periodic case given above. During → ∞ this transition the resulting basis of plane waves should be normalized to a δ function. This is based on the fact that k =2πn/L, causing the k-spectrum to become very closely spaced for L and the summation over n to become an integral over k: →∞

∞ L L ∞ →∞ dk . (A.6) −−→ 2π n= Z X−∞ −∞ The relevant basis functions are the momentum eigenfunctions belonging to the momentum eigenvalues ~k:

L (A.1) 1 fk(x) = fn(x) ==== exp(ikx) (k IR) , (A.7) r2π √2π ∈ with orthonormality relation

1 ∞ dx exp ix[k k′] = δ(k k′) . (A.8) 2π − − Z −∞ The Fourier series now becomes a Fourier transformation (Fourier integral):

1 ∞ 1 ∞ f(x) = dk A(k) exp(ikx) and A(k) = dx f(x) exp( ikx) . √2π √2π − Z Z −∞ −∞ (A.9)

ii A.2.1 Deﬁnition of the δ function

The δ function in equation (A.8) is deﬁned on an arbitrary continuous test function g(x):

∞ g(x) = dx′ g(x′)δ(x x′) . (A.10) ∀ − g Z −∞

Since g(x) is arbitrary, δ(x x′) should vanish for x = x′ . The peak of δ(x x′) at − 6 − x = x′ should be such that the integral over δ(x x′) around x = x′ always yields one. − The δ function therefore only exists as a limit of a series of functions that progressively become more peaked around x = x′. The result of such a limiting procedure is called a distribution. An example of this is given by the integral

1 ∞ 1 1 1 1 ǫ δ (x) = dk exp(ikx ǫ k ) = = (ǫ> 0) . ǫ 2π − | | 2π ix + ǫ − ix ǫ π x2 + ǫ2 Z − −∞ For decreasing ǫ the function δ (x) becomes smaller for all x = 0, whereas simultaneously ǫ 6

x2 >0 x2/ǫ v =x/ǫ 1 dv 1 ǫ 0 dx δ (x) ==== = arctan(x /ǫ) arctan(x /ǫ) ↓ 1 . ǫ π v2 +1 π 2 − 1 −−→ x1Z<0 x1Z/ǫ The following series representations of the δ function will be used in these lecture notes:

1 ǫ 1 ∞ δ(x) = lim = dk exp(ikx) , (A.11) ǫ 0 π x2 + ǫ2 2π ↓ Z −∞ θ(x + ǫ) θ(x) δ(x) = lim − = θ′(x) , (A.12) ǫ 0 ǫ ↓ where θ(x) is the usual Heaviside step function.

The following handy identities hold for the δ function:

g(x)δ(x c) = g(c)δ(x c) , (A.13) − − as well as 1 δ h(x) = δ(x x ) for h(x ) = 0 and h′(x ) =0 , (A.14) h (x ) − j j j 6 j | ′ j | X provided that h(x) is suﬃciently well-behaved close to the roots of the equation h(x) = 0.

In three dimensions, ﬁnally, the following compact notation can be used:

∞ ∞ ∞ 1 (A.11) δ(~r ) δ(x)δ(y)δ(z) dk dk dk exp(i~k ~r ) ==== δ(~r ) . (A.15) ≡ ⇒ (2π)3 x y z · Z Z Z −∞ −∞ −∞

iii B Properties of the Pauli spin matrices

Since the world of QM is littered with spin-1/2 particles (such as electrons, nucleons, etc.), we give here a short summary of the properties of the Pauli spin matrices

0 1 0 i 1 0 σx = , σy = − and σz = . (B.1) 1 0 i 0 0 1 ! ! − ! ˆ These 2 2 matrices are linked directly to the spin operator S~ for spin-1/2 particles: × 2 ˆ ~σ S.~ (B.2) ≡ ~

From the fundamental commutator algebra for the spin operator and the fact that Sˆz has eigenvalues ~/2, the following properties of the Pauli spin matrices can be derived: ±

σ = σ† , Tr(σ ) = 0 , det(σ ) = 1 (k = x, y, z) k k k k −

and σx, σy = 2iσz (cyclic) . (B.3) Denoting the identity matrix in spin space by I2, the following additional identities hold for the Pauli spin matrices:

σ2 = I (k = x, y, z) and σ σ = σ σ = iσ (cyclic) k 2 x y − y x z σ , σ = 2δ I (j,k = x, y, z ) . (B.4) ⇒ j k jk 2 This allows us to complete the relevant trace identities in spin space to

(B.3) (B.4) Tr(I2) =2 , Tr(σj) ==== 0 and Tr(σjσk) ==== 2δjk (j,k = x, y, z ) . (B.5) Moreover, for arbitrary 3-dimensional vectors A~ and B~ the following holds:

1 (B.3),(B.4) (~σ A~ )(~σ B~ ) = A B σ , σ + σ , σ ======(A~ B~ )I + i~σ (A~ B~ ) , · · 2 j k j k j k · 2 · × j,k X (B.6) using that

A B (B.3) j k σ , σ ==== iσ (A B A B )+ iσ (A B A B )+ iσ (A B A B ) . 2 j k z x y − y x x y z − z y y z x − x z j,k X Combined with the identity matrix the Pauli spin matrices form a basis of 2 2 matrices. × As such, an arbitrary 2 2 matrix A can be decomposed as × A = A I + A~ ~σ (A C and A C) . (B.7) 0 2 · 0 ∈ x,y,z ∈

iv C Lagrange-multiplier method

Suppose we want to determine the extrema of a quantity F (q , , q ) while simultane- 1 ··· N ously trying to satisfy the boundary condition f(q , , q ) = C . In that case it makes 1 ··· N good sense to employ the so-called Lagrange-multiplier method. This consists in looking for solutions to the variational equation

δ F (q , , q ) λ(q , , q ) f(q , , q ) C = 0 , (C.1) 1 ··· N − 1 ··· N 1 ··· N − where δ denotes the action of independently varying the coordinates q , , q as well as 1 ··· N the Lagrange multiplier λ. This results in

N ∂F ∂λ ∂f (f C) λ δq (f C) δλ = 0 ∂q − − ∂q − ∂q j − − ∀ j =1 j j j δλ,δqj X ∂F ∂f f(q1, , qN ) = C and = λ (j =1, , N ) . (C.2) ⇒ ··· ∂qj ∂qj ··· In this way the boundary condition is indeed satisﬁed automatically. By substituting the obtained solution(s) for q , , q in the boundary condition f(q , , q ) = C , the 1 ··· N 1 ··· N Lagrange multiplier λ can be related to the constant C . The Lagrange-multiplier method is used in these lecture notes to maximize the entropy of ensembles in thermal equilibrium while simultaneously imposing certain additional constraints.

D Special relativity: conventions and deﬁnitions

In this appendix we specify a few conventions that will allow us to deal with ﬂat spacetime (Minkowski space). The time coordinate t and spatial coordinates ~r of a particle are combined into a contravariant position 4-vector

x1 xµ : x0 ct and x2 ~x ~r , (D.1) ≡   ≡ ≡ x3   with µ = 0, 1, 2, 3 the Minkowski index. Also the energy E and momentum ~p are combined into a contravariant momentum 4-vector

1 mc p m~v pµ : p0 E/c = and p2 ~p = , (D.2) 2 2 2 2 ≡ 1 ~v /c  3  ≡ 1 ~v /c − p − p   p corresponding to a particle with rest mass m and velocity ~v = d~r/dt.

v The metric tensor in Minkowski space is given by

1 Ø g = gµν = 1 . (D.3) µν  − 1  Ø − 1  −  µ   From the contravariant vector a the corresponding covariant vector aµ can be derived as

a g aν . (D.4) µ ≡ µν Here we made use of the Einstein summation convention, which states that a repeated Minkowski index automatically implies summation of that index. Finally we introduce the inner product (scalar product)

x y xµy = x yµ = g xµyν = x0y0 ~x ~y, (D.5) · ≡ µ µ µν − · which is not positive deﬁnite as a result of the spacetime metric gµν in Minkowski space.

Inertial systems: reference frames in which force-free particles travel along straight lines are referred to as inertial systems. Relativity principle: all inertial systems are physically equivalent and the speed of light is the same in each inertial system.

The relativity principle is reﬂected in the transformation characteristic of position vectors

while going from inertial system S with origin O to inertial system S′ with origin O′.

Assume the origins to coincide (i.e. O = O′ ) at time t = 0 and indicate the corresponding coordinate transformation as follows:

µ µ µ ν S : x S′ : x′ = Λ x . (D.6) → ν The coordinate transformation (Lorentz transformation) Λ has to be linear to be able to map linear motions on linear motions in the absence of forces. It can be deﬁned in accordance with the relativity principle by demanding x2 x x to be invariant under the ≡ · transformation. After all, in that case a particle that moves at the speed of light starting from the origin O at t = 0, which implies c2 = ~v 2 = ~x 2/t2 x2 = 0, automatically ⇒ 2 2 travels with the speed of light in any other inertial system S′, since x′ = x = 0 implies 2 2 2 2 ~v ′ = ~x ′ /t′ = c . From this condition we can derive that

2 2 (D.5),(D.6) ρ σ µ ν ρ σ µ x = x′ ======gρσ x x = Λ ρΛ σ gµν x x Λ ρΛµσ = gρσ , (D.7) x∀ ⇒x ∀ ⇒ which implies that (Λ0 )2 = 1+(Λ1 )2 + (Λ2 )2 + (Λ3 )2 1. The inverse of the Lorentz 0 0 0 0 ≥ transformation then becomes

1 Ø µρ ρ 1 µρ 1 ρµ Λ Λ = g = Λ = (Λ− ) , det(Λ) = 1 . (D.8) µσ σ  1  ⇒ ± Ø 1     vi The Lorentz transformations can be subdivided as follows. First of all there are the so- called proper orthochronous Lorentz transformations (or short: “the Lorentz transformations”), for which det(Λ) = +1 and Λ0 1. These transformations comprise: 0 ≥ spatial rotations, about three independent spatial axes. •

Lorentz boosts, in three independent spatial directions. Example: S′ moves relative • to S at constant velocity v = βc in the x1-direction. In that case

1/ 1 β2 β/ 1 β2 0 0 cosh η sinh η 0 0 − − − 2 2 − µ  β/p1 β 1/p1 β 0 0   sinh η cosh η 0 0  Λ ν = − − − − (D.9) p0p 0 10 ≡ 0 0 10      0 0 01   0 0 01          1 0 0 0 β 1,x0=ct ≪ non-relativistic Galilei transformation : v/c 1 0 0 . −−−−−−→  − 0 0 1 0  0 0 0 1     Two more classes of Lorentz transformations can be added to this: transformations with det(Λ) = 1 and Λ0 1, such as the parity transformation that describes spatial inver- − 0 ≥ sion, and transformations with Λ0 1, such as the time-reversal transformation. All 0 ≤ − transformations can be combined into the so-called homogeneous Lorentz transformations, for which an arbitrary transformation can be written as a product of rotations, boosts, spatial inversion and time reversal. Spatial inversion and time reversal have a special status:

free-particle systems are invariant under these transformations, but this invariance • gets lost once the weak (nuclear) force enters the game;

these transformations cannot be connected continuously with the identity transfor- • mation, as opposed to the proper orthochronous Lorentz transformations which form a continuous group and for which a finite transformation can be written as an infinite series of infinitesimal transformations. Spatial inversion and time reversal in fact incorporate the possible discrete Lorentz transformations.

Finally, we also have to consider the possibility that the origin O′ is shifted with respect to the origin O:

µ µ ν µ µ x′ = Λ ν x + a (short: x′ =Λx + a) , with a a constant 4-vector

µ 1 µ ν 1 x = (Λ− ) (x′ a) (short: x =Λ− [x′ a]) . (D.10) ⇒ ν − − All transformations together form the so-called inhomogeneous Lorentz transformations, which are also known as the Poincarétransformations. They are defined by demanding 2 2 the invariance (x y) =(x′ y′ ) . − − vii For the construction of a relativistic wave equation, which should have the same form in all inertial systems, we need several types of quantities (independent of space and time) and fields (dependent on space and time).

Scalar quantities: quantities that remain the same in each inertial system, such as • p2 = p p and the speed of light c. · Scalar ﬁelds: ﬁelds with the following transformation characteristic • (D.10) 1 S : φ(x) S′ : φ′(x′) = φ(x) ==== φ Λ− [x′ a] . → − µ µ ν Contravariant vector quantities: 4-vectors that transform as A′ = Λ A , such as • ν the momentum 4-vector pµ.

Contravariant vector ﬁelds: ﬁelds with the following transformation characteristic • µ µ µ ν (D.10) µ ν 1 S : A (x) S′ : A′ (x′)=Λ A (x) ==== Λ A Λ− [x′ a] . → ν ν − Covariant vector quantities: 4-vectors that transform as • ρ ρ σ ρ σν ν (D.8) 1 ν A′µ = gµρ A′ = gµρ Λ σA = gµρ Λ σ g Aν = Λµ Aν ==== (Λ− ) µ Aν .

Covariant vector ﬁelds : ﬁelds with the following transformation characteristic • ρ ρ σ ρ σν S : A (x) S′ : A′ (x′) = g A′ (x′) = g Λ A (x) = g Λ g A (x) µ → µ µρ µρ σ µρ σ ν

(D.10) ν 1 (D.8) 1 ν 1 ==== Λ A Λ− [x′ a] ==== (Λ− ) A Λ− [x′ a] . µ ν − µ ν − Examples: the gradient in Minkowski space, ∂ ∂ 1 ∂ 1 ∂ ∂ = and ∂ = ~ , (D.11) µ ≡ ∂xµ ⇒ 0 c ∂t  2  ▽ ∂3 is a covariant vector quantity since  

ν ∂ ∂x ∂ (D.10) 1 ν ∂ (D.8) ν ∂ ν ′ − ∂µ = µ = µ ν ==== (Λ ) µ ν ==== Λµ ν = Λµ ∂ν . ∂x′ ∂x′ ∂x ∂x ∂x µ µν Consequently, ∂ = g ∂ν is automatically a contravariant vector quantity. Furthermore

A (x) ∂ φ(x) is a covariant vector ﬁeld if φ(x) is a scalar ﬁeld. • µ ≡ µ ν ν Proof: Aµ′ (x′) = ∂µ′ φ′(x′)=Λµ ∂ν φ(x)=Λµ Aν(x).

φ(x) A (x)Bµ(x) is a scalar ﬁeld if A and B are vector ﬁelds. • ≡ µ µ 1 ν µ ρ ν Proof: φ′(x′) = Aµ′ (x′)B′ (x′) = Aν(x)(Λ− ) µ Λ ρ B (x) = Aν (x)B (x)= φ(x).

φ(x) ∂ Aµ(x) is a scalar ﬁeld if Aµ(x) is a contravariant vector ﬁeld (see previous • ≡ µ example).

viii E Electromagnetic orthonormality relations

Consider the periodic electromagnetic Fourier components

1 ~ ~u~ (~r ) = ~ǫλ(~ek) exp(ik ~r ) , (E.1) k,λ √V · where the wave vector ~k is allowed to take on the following quantized values: ω 2π ~k k ~e k ~e = ~ν, with ν = 0, 1, 2, ~ν = ~0 . (E.2) ≡ k ≡ c k L x,y,z ± ± ··· ∧ 6

Together with the propagation direction ~ek the polarisation vectors ~ǫ1(~ek) and ~ǫ2(~ek) form an orthonormal right-handed coordinate system:

~ǫ (~e ) ~ǫ (~e ) = ~e , ~ǫ (~e ) ~e = ~ǫ (~e ) , ~e ~ǫ (~e ) = ~ǫ (~e ) , 1 k × 2 k k 2 k × k 1 k k × 1 k 2 k

~ǫ (~e ) ~ǫ ′ (~e ) = δ ′ . (E.3) λ k · λ k λ,λ By means of the orthonormality relation for periodic Fourier modes: 1 d~r exp i~r [~k ~k′ ] = δ ′ , (E.4) V · − ~k,~k Z V the following identities can be derived for the Fourier components:

(E.4) (E.3) ′ ′ d~r~u (~r ) ~u ∗′ ′ (~r ) ==== ~ǫ (~e ) ~ǫ (~e ) δ ′ ==== δ δ ′ , (E.5) ~k,λ · ~k ,λ λ k · λ k ~k,~k λ,λ ~k,~k VZ

(E.4) ′ ′ d~r ~e ~u (~r ) ~e ~u ∗′ ′ (~r ) ==== ~e ~ǫ (~e ) ~e ~ǫ (~e ) δ ′ k × ~k,λ · k × ~k ,λ k × λ k · k × λ k ~k,~k VZ (E.3) ′ ==== δλ,λ δ~k,~k′ , (E.6)

(E.4) ′ ′ d~r~u (~r ) ~e ~u ∗′ ′ (~r ) ==== ~ǫ (~e ) ~e ~ǫ (~e ) δ ′ ~k,λ × k × ~k ,λ λ k × k × λ k ~k,~k VZ (E.3) ′ ==== ~ek δλ,λ δ~k,~k′ , (E.7)

(E.4) ′ d~r~u (~r ) ~u ∗′ ′ (~r ) ==== ~ǫ (~e ) ~ǫ (~e ) δ ′ ~k,λ × ~k ,λ λ k × λ k ~k,~k VZ

(E.3) ==== ~e (δ δ ′ δ δ ′ ) δ ′ . (E.8) k λ,1 λ ,2 − λ,2 λ ,1 ~k,~k

ix Index γ-matrices of Dirac, 112 continuum limit, 71, 76 for the e.m. field, 134 absorption, 137 contravariant vector fields, viii additive many-particle quantities contravariant vector quantities, viii 1-particle quantities, 16 Cooper instability, 46 2-particle quantities, 18 Coulomb conditions, 125 annihilation operator, 7 Coulomb gauge, 125 anticommutation relations, 10, 13 covariant vector fields, viii anticommutator, 10 covariant vector quantities, viii atomic heat bath, 142 creation operator, 6 average occupation numbers, 88 Curie’s law, 65 binding energy, 75 d’Alembertian, 104 for nuclei, 82 Darwin term, 120 black body, 142 de Broglie hypothesis, 103 Bogolyubov transformation, 38 density matrix, 50, 53 bosonic, 39 density of states, 71 fermionic, 47 for photons, 132 Boltzmann constant, 60 density operator, 53 Bose gas, 93 equation of motion, 59 degenerate, 94 dipole selection rules, 141 Bose–Einstein condensation, 90, 94 Dirac equation, 109 critical temperature, 93 anticommutation relations, 150 Bose–Einstein statistics, 2, 87 Dirac spinors, 110 bosons, 2, 10 energy density, 149 box, 69 field operator, 150 Casimir effect, 133 spin-1/2 antiparticles, 151 Chandrasekhar limit, 86 spin-1/2 particles, 115, 151 charge conjugation, 144 total charge operator, 151 charge number, 80 Dirac representation, 111 chemical potential, 68, 99 Dirac sea, 116 classical gas, 92 Dirac spin operator, 115 coherent state, 27 distinguishable, 1 commutation relations, 10, 13 electric dipole approximation, 140 Compton wavelength, 122 electromagnetism conduction electrons, 77 4-current, 124 continuity equation, 106, 111 4-vector potential, 118, 124

x classical limit, 138 Fourier integral, ii commutation relations, 129 Fourier modes, i field operator, 129 Fourier series, i spin-1 particles, 130 Fourier transformation, ii electron gas, 77 freezing out degrees of freedom, 90 electron–hole excitation, 45 fugacity, 68 emission, 137 g-factor, 120 spontaneous, 138 gauge freedom, 125 lifetime, 141 stimulated, 138 heat bath, 62 energy quanta, 23, 25 heat capacity, 92 photon, 130 Heisenberg equation of motion, 21 ensemble average, 53 Heisenberg picture, 21 time evolution, 59 helicity, 118, 130 ensembles helium film, 44 canonical, 62 helium I, 42 completely random, 58, 59 helium II, 42 grand canonical, 67 hole, 47 microcanonical, 65 hole theory, 45 mixed, 51, 57 pure, 49, 57, 59 ideal gas, 3, 90 unpolarized, see completely random identical particles, 1 entropy, 60 incoherent mixture, 51 equipartition of energy, 64 indistinguishable, 1 inertial system, vi Fermi energy, 45, 74, 76, 89 intrinsic degrees of freedom, 105 Fermi gas, 69, 98 intrinsic magnetic dipole moment, 120 completely degenerate, 74, 99 isolated system, 62 degenerate, 98 Fermi momentum, 76 Klein–Gordon equation, 104 Fermi sea, 45, 74 commutation relations, 148 Fermi sphere, 76 energy density, 146 Fermi surface, 76 scalar field, 106 Fermi temperature, 75 scalar field operator, 147 Fermi–Dirac statistics, 2, 87 spin-0 antiparticles, 147 fermions, 2, 10 spin-0 particles, 106, 147 field operator, 15 total charge operator, 148 fine structure, 121 Lagrange multiplier, v Fock space, 5 Lagrange-multiplier method, v fontain effect, 43

xi law of Stefan–Boltzmann, 142 occupation-number representation, 6 left-handed circularly polarized light, 131 order parameter, 96 Liouville equation, 59 oscillators locality, 148 bosonic quanta, 25, 27 long-range correlations, 96 forced, 26 long-range order, 96 free, 23 Lorentz transformations, vi overlapping wave functions, 1 homogeneous, vii pair number operator, 20 Lorentz boosts, vii paramagnetic susceptibility, 65, 101 orthochronous, vii particle interpretation, 23 proper Lorentz transformations, vii particle–wave duality, 103, 145 spatial rotations, vii partition function lowering operator, 24 canonical, 63 macroscopically occupied, 94 grand canonical, 68 magnetization, 65 Pauli equation, 120 magnetization density, 100 Pauli exclusion principle, 5, 11 mass number, 80 Pauli spin matrices, iv, 56 Maxwell equations, 123 periodic plane waves, i Maxwell–Boltzmann statistics, 87 permutation degeneracy, 1 microcausality, 149 permutation operator, 2 minimal substitution, 118, 135 persistent current, 43 Minkowski space, v photon gas, 142 contravariant vector, v photon states, 131 covariant vector, vi photon statistics, 142 Einstein summation convention, vi photons, 130 inner product, vi Planck’s law, 142 metric tensor, v plasma, 77 Minkowski index, v Poincar´etransformations, vii polarization for photons neutron gas, 87 polarization vector, 126 neutron star, 87 polarization for spin-1/2 particles non-interacting many-particle systems, 3 degree of polarization, 57 normal ordering, 19, 132 direction of polarization, 57 nucleon distribution, 80 polarization vector, 56 number density operator, 14 pressure ionization, 84 number operators, 6, 7 pure state, 49, 54 observables for identical particles, 1 quasi particles, 26, 27, 38, 41 occupation numbers, 6 quasi-particle vacuum, 27 distributions, 88

xii raising operator, 24 white dwarf, 84 relativistic quantum field theory, 145 zero-point energy, 25, 132 relativity principle, vi, 103 reservoir, 66 right-handed circularly polarized light, 131 Rollin film, 44 saturation, 83 scalar fields, viii scalar potential, 123 scalar quantities, viii Schrödinger field, 22 Schrödinger picture, 21 second quantization, 18, 22, 135, 144–152 slash notation, 112 Slater determinant, 4 spin–orbit interaction, 120 spin–statistics theorem, 152 state functions product functions, 4 totally antisymmetric, 2, 14 totally symmetric, 2, 14 statistical weight, 53 super leak, 43 superfluidity, 31 critical velocity, 32 Landau criterion, 32 symmetrization postulate, 1 temperature, 63 thermal de Broglie wavelength, 91 thermal equilibrium, 61 Thomas–Fermi model, 80 total number operator, 8, 14 transversality, 126 two-fluid model, 42 vacuum state, 6 vector potential, 123 wave vector, 71, 76, 126

xiii