2.2 Quantum Mechanics and First Quantization

Home , First quantization

2.2. QUANTUM MECHANICS AND FIRST QUANTIZATION 65 explained by Maxwell’s theory of the electromagnetic field. The classical expression for the field energy of an electromagnetic wave which only depends on the field amplitudes E0, B0 and, therefore, is a continuous function, cf. hamiltonian (2.21), did not reproduce the measured spectral energy density. The solution which was found by Max Planck indicated that the field energy cannot depend on the field amplitudes alone. The energy exchange between electromagnetic field and matter is even entirely independent of E0, B0, instead it depends on the frequency ω of the wave. Thus the total energy of an electromagnetic wave of frequency ω is an integer multiple of an elementary energy, Wfield(ω)= N~ω, where ~ is Planck’s constant, i.e. the energy is quantized. What Planck had discovered in 1900 was the quantization of the electromagnetic field 4. This concept is very different from the quantum mechanical description of the electron dynamics from which it is, therefore, clearly distin- guished by the now common notion of “second quantization”. Interestingly, however, the “first quantization” of the motion of microparticles was introduced only a quarter century later when quantum mechanics was discovered.

2.2 Quantum mechanics and ﬁrst quantization

2.2.1 Reminder: State vectors and operators in Hilbert space Let us brieﬂy recall the main ideas of quantum theory. The essence of quantum mechanics or “ﬁrst” quantization is to replace functions by operators, starting from the coordinate and momentum (here we use the momentum representation), r ˆr = r, → ~ ∂ p pˆ = , → i ∂r where the last equalities refer to the coordinate representation. These operators are hermitean, ˆr† = ˆr and pˆ† = pˆ, and do not commute

[ˆxi, pˆj]= i~δi,j, (2.23) which means that coordinate and momentum (the same components) cannot be measured simultaneously. The minimal uncertainty of such a simultaneous

4Planck himself regarded the introduction of the energy quantum ~ω only as a formal mathematical trick and did not question the validity of Maxwell’s field theory. Only half a century later, when field quantization was systematically derived, the coexistence of the concepts of energy quanta and electromagnetic waves became understandable, see Sec. 1.5 66 CHAPTER 2. SECOND QUANTIZATION measurement is given by the Heisenberg relation ~ ∆ˆx ∆ˆp , (2.24) i i ≥ 2 where the standard deviation (“uncertainty”) of an operator Aˆ is definied as

2 ∆Aˆ = Aˆ Aˆ , (2.25) s −h i and the average is computed in a given state ψ , i.e. Aˆ = ψ Aˆ ψ . The general formulation of quantum mechanics describes| i anh arbii traryh | | quantumi system in terms of abstract states ψ that belong to a Hilbert space (Dirac’s notation), and operators act on these| i state returning another Hilbert space state, Aˆ ψ = φ . The central| i | quantityi of classical mechanics – the hamilton function – retains its functional dependence on coordinate and momentum in quantum mechanics as well (correspondence principle) but becomes an operator depending on operators, H(r, p) Hˆ (ˆr, pˆ). The classical equations of motion – Hamil- ton’s equations or Newton’s→ equation (2.4) – are now replaced by a partial differential equation for the Hamilton operator, the Schrödinger equation ∂ i~ ψ(t) = Hˆ ψ(t) . (2.26) ∂t| i | i Stationary properties are governed by the stationary Schrödinger equation that follows from the ansatz5

i ˆ ψ(t) = e− ~ Ht ψ , | i | i Hˆ ψ = E ψ . (2.27) | i | i The latter is an eigenvalue equation for the Hamilton operator with the eigenfunctions ψ and corresponding eigenvalues E. | i 2.2.2 Probabilistic character of “First” quantization. Comparison to experiments Experiments in quantum mechanics never directly yield the wave function or the probability distribution. Individual (random) realizations of possible conﬁgurations and their dynamics. Examples:

5Here we assume a time-independent hamiltonian. 2.3. THE LADDER OPERATORS 67

1. double slit experiment with electrons of Tonomura

2. photons on photo plate,

3. many-body dynamics in ultracold atom experiments in optical lattices

Possible configurations in particular evident in the case of fermionic atoms (assuming spin s = 1/2, for simplicity) in optical lattices. Due to the Pauli principle each lattice site can be occupied only by zero, one or two atoms – in the latter case they have to have different spin projections. If an initial config- uration of atoms is excited (e.g. by a confinement quench), a dynamical evo- lution will start. This will, of course, not be described by the time-dependent Schrödinger equation for the many-atom wave function! This equation only describes the average dynamics that follow from averaging over the dynamics that start from many independent realizations. This has been very successful but, on the other hand, reproduces only part of the information. For example, it completely misses the fluctuations of the numbers of atoms around the average.6 “First” quantization is evident in the case of particle motion in a confining potential U(r), such as an oscillator potential: classical bounded motion transforms, in quantum mechanics, into a set of eigenstates ψ (that are localized | ni as well) that exist only for a sequence of discrete (quantized) energies En. This example is discussed more in detail below.

2.3 The linear harmonic oscillator and the ladder operators

Let us now recall the simplest example of quantum mechanics: one particle in m 2 2 a one-dimensional harmonic potential U(x)= 2 ω x , i.e. in Eq. (2.1), N =1 and the interaction potentials vanish. We will use this example to introduce the basic idea of “second quantization”. In writing the potential U(x) we switched to the coordinate representation where states ψ are represented by | ni functions of the coordinate, ψn(x). At the end we will return to the abstract notation in terms of Dirac states.

6this section is not complete yet. 68 CHAPTER 2. SECOND QUANTIZATION

2.3.1 One-dimensional harmonic oscillator The stationary properties of the harmonic oscillator follow from the stationary Schr¨odinger equation (2.27) which now becomes, in coordinate representation

pˆ2 mω2 Hˆ (ˆx, pˆ)ψ (x)= + xˆ2 ψ (x)= E ψ (x), (2.28) n 2m 2 n n n ~ d wherex ˆ = x andp ˆ = i dx . We may bring the Hamilton operator to a more symmetric form by introducing the dimensionless coordinate ξ = x/x0 with 1/2 the length scale x0 =[~/mω] , whereas energies will be measured in units of ~ω. Then we can replace d = 1 d and obtain dx x0 dξ

Hˆ 1 ∂2 = + ξ2 . (2.29) ~ω 2 −∂ξ2 This quadratic form can be rewritten in terms of a product of two ﬁrst order operators a, a†, the “ladder operators”,

1 ∂ a = + ξ , (2.30) √2 ∂ξ 1 ∂ a† = + ξ . (2.31) √2 −∂ξ Indeed, computing the product

1 ∂ 1 ∂ Nˆ = a†a = + ξ + ξ (2.32) √2 ∂ξ √2 −∂ξ 1 ∂2 = + ξ2 1 , 2 −∂ξ2 − the hamiltonian (2.29) can be written as

Hˆ 1 = Nˆ + . (2.33) ~ω 2

It is obvious from (2.33) that Nˆ commutes with the hamiltonian,

[H,ˆ Nˆ]=0, (2.34) and thus the two have common eigenstates. This way we have transformed the hamiltonian from a function of the two non-commuting hermitean operators 2.3. THE LADDER OPERATORS 69 xˆ andp ˆ into a function of the two operators a and a† which are also non- commuting7, but not hermitean, instead they are the hermitean conjugate of each other,

[a, a†] = 1, (2.35) (a)† = a†, (2.36) which is easily veriﬁed. The advantage of the ladder operators is that they allow for a straightforward computation of the energy spectrum of Hˆ , using only the properties (2.32) and (2.35), without need to solve the Schr¨odinger equation, i.e. avoid- 8 ing explicit computation of the eigenfunctions ψn(ξ) . This allows us to return to a representation-independent notation for the eigenstates, ψn n . The only thing we require is that these states are complete and orthonormal,→ | i 1= n n and n n′ = δ ′ . n | ih | h | i n,n Now, acting with Nˆ on an eigenstate, using Eq. (2.33), we obtain P E 1 Nˆ n = a†a n = n n = n n , n, (2.37) | i | i ~ω − 2 | i | i ∀ E 1 n = n , ~ω − 2 where the last line relates the eigenvalues of Nˆ andn ˆ that correspond to the common eigenstate n . Let us now introduce two new states that are created by the action of the| ladderi operators,

a n = n˜ , | i | i a† n = n¯ , | i | i where this action is easily computed. In fact, multiplying Eq. (2.37) from the left by a, we obtain E 1 aa† n˜ = n n˜ . | i ~ω − 2 | i Using the commutation relation (2.35) this expression becomes E 3 a†a n˜ = n n˜ =(n 1) n˜ , | i ~ω − 2 | i − | i 7The appearance of the standard commutator indicates that these operators describe bosonic excitations. 8We will use the previous notation ψ, which means that the normalization is x dξ ψ(ξ) 2=1 0 | | R 70 CHAPTER 2. SECOND QUANTIZATION which means the state n˜ is an eigenstate of Nˆ [and, therefore, of Hˆ ] and has an energy lower than n| byi ~ω whereas the eigenvalue of Nˆ isn ˜ = n 1. Thus, the action of the operator| i a is to switch from an eigen state with− eigenvalue n to one with eigenvalue n 1. Obviously, this is impossible for the ground state, i.e. when a acts on 0−, so we have to require | i 0˜ = a 0 0. (2.38) | i | i≡ When we use this result in Eq. (2.37) for n = 0, the l.h.s. is zero with the consequence that the term in parantheses must vanish. This immediately leads to the well-known result for the ground state energy: E0 = ~ω/2, corresponding to the eigenvalue 0 of Nˆ. From this we now obtain the energy spectrum of the excited states: acting with a† from the left on Eq. (2.37) and using the commutation relation (2.35), we obtain E 1 Nˆ n¯ = n +1 n¯ =n ¯ n¯ . | i ~ω − 2 | i | i Thus,n ¯ is again an eigenstate of Nˆ and Hˆ . Further, if the eigenstate n | i has an energy En, cf. Eq. (2.37), thenn ¯ has an energy En + ~ω, whereas the associated eigenvalue of Nˆ isn ¯ = n + 1. Starting from the ground state † and acting repeatedly with a we construct the whole spectrum, En, and may express all eigenfunctions via ψ0:

1 E = ~ω n + , n =0, 1, 2,... (2.39) n 2 n n = C a† 0 . (2.40) | i n | i 1 Cn = , (2.41) √n!

† where the normalization constant Cn will be veriﬁed from the properties of a below. The above result shows that the eigenvalue of the operator Nˆ is just the quantum number n of the eigenfunction n . In other words, since n is obtained by applying a† to the ground state| functioni n times or by “n-fold| i excitation”, the operator Nˆ is the number operator counting the number of excitations (above the ground state). Therefore, if we are not interested in the analytical details of the eigenstates we may use the operator Nˆ to count the number of excitations “contained” in the system. For this reason, the common notion for the operator a (a†) is “annihilation” (“creation”) operator of an excitation. For an illustration, see Fig. 2.3.1. 2.3. THE LADDER OPERATORS 71

Figure 2.1: Left: oscillator potential and energy spectrum. The action of the operators a and a† is illustrated. Right: alternative interpretation: the operators transform between “many-particle” states containing diﬀerent number of elementary excitations.

From the eigenvalue problem of Nˆ, Eq. (2.37) we may also obtain the explicit action of the two operators a and a†. Since the operator a transforms a state into one with quantum number n lower by 1 we have

a n = √n n 1 , n =0, 1, 2,... (2.42) | i | − i where the prefactor may be understood as an ansatz9. The correctness is proven by deriving, from Eq. (2.42), the action of a† and then verifying that we recover the eigenvalue problem of Nˆ, Eq. (2.37). The action of the creation operator is readily obtained using the property (2.36):

a† n = n¯ n¯ a† n = n¯ a[ n¯ ] n | i | ih | | i | i h | | i n¯ n¯ X X = n¯ √n¯ n¯ 1 n = √n +1 n +1 . (2.43) | i h − | i | i n¯ X Inserting these explicit results for a and a† into Eq. (2.37), we immediately verify the consistency of the choice (2.42). Obviously the oscillator eigenstates n are no eigenstates of the creation and annihilation operators 10. | i Problems:

1. Calculate the explicit form of the ground state wave function by using Eq. (2.38).

2. Show that the matrix elements of a† are given by n +1 a† n = √n + 1, where n =0, 1,... , and are zero otherwise. | |

3. Show that the matrix elements of a are given by n 1 a n = √n, where n =0, 1,... , and are zero otherwise. h − | | i

4. Proof relation (2.41). 9This expression is valid also for n = 0 where the prefactor assures that application of a to the ground state does not lead to a contradiction. 10A particular case are Glauber states (coherent states) that are a special superposition of the oscillator states which are the eigenstate of the operator a. 72 CHAPTER 2. SECOND QUANTIZATION

2.3.2 Generalization to several uncoupled oscillators

The previous results are directly generalized to a three-dimensional harmonic oscillator with frequencies ωi, i =1, 2, 3, which is described by the hamiltonian

Hˆ = Hˆ (ˆxi, pˆi), (2.44) i=1 X which is the sum of three one-dimensional hamiltonians (2.28) with the po- m 2 2 2 2 2 2 tential energy U(x1,x2,x3) = 2 (ω1x1 + ω2x2 + ω3x3). Since [pi,xk] δk,i all three hamiltonians commute and have joint eigenfunction (product states).∼ The problem reduces to a superposition of three independent one-dimensional oscillators. Thus we may introduce ladder operators for each component inde- pendently as in the 1d case before,

1 ∂ ai = + ξi , (2.45) √2 ∂ξi † 1 ∂ † ai = + ξi , [ai, ak]= δi,k. (2.46) √2 −∂ξi

Thus the hamiltonian and its eigenfunctions and eigenvalues can be written as

3 1 Hˆ = ~ω a†a + i i i 2 i=1 X a 0 = 0, i =1, 2, 3 i| i 1 † n1 † n2 † n3 ψn1,n2,n3 = n1n2n3 = (a1) (a2) (a3) 0 (2.47) | i √n1!n2!n3! | i 3 1 E = ~ω n + . i i 2 i=1 X

Here 0 000 = 0 0 0 denotes the ground state and a general state | i≡| i | i| i| i n1n2n3 = n1 n2 n3 contains ni elementary excitations in direction i, cre- | i | i| i| i † ated by ni times applying operator ai to the ground state. Finally, we may consider a more general situation of any number M of coupled independent linear oscillators and generalize all results by replacing the dimension 3 M. → 2.4. INTERACTING PARTICLES 73

Figure 2.2: Illustration of the one-dimensional chain with nearest neighbor interaction. The chain is made inﬁnite by connecting particle N + 1 with particle 1 (periodic boundary conditions).

2.4 Generalization to interacting particles. Normal modes

The previous examples of independent linear harmonic oscillators are of course the simplest situations which, however, are of limited interest. In most problems of many-particle physics the interaction between the particles which was neglected so far, is of crucial importance. We now discuss how to apply the formalism of the creation and annihilation operators to interacting systems.

2.4.1 One-dimensional chain and its normal modes We consider the simplest case of an interacting many-particle system: N identical classical particles arranged in a linear chain and interacting with their left and right neighbor via springs with constant k.11, see Fig. 2.2.

This is the simplest model of interacting particles because each particle is assumed to be ﬁxed around a certain position xi in space around which it can perform oscillations with the displacement qi and the associated momentum 12 pi. Then the hamiltonian (2.1) becomes

N p2 k H(p,q)= j + (q q )2 . (2.48) 2m 2 j − j+1 j=1 X Applying Hamilton’s equations we obtain the system of equations of motion (2.4) mq¨ = k (q 2q + q ) , j =1 ...N (2.49) j j+1 − j j−1 which have to be supplemented with boundary and initial conditions. In the following we consider a macroscopic system and will not be interested in ef- fects of the left and right boundary. This can be achieved by using “periodic” boundary conditions, i.e. periodically repeating the system according to

11Here we follow the discussion of Huang [Hua98]. 12Such “lattice” models are very popular in theoretical physics because they allow to study many-body eﬀects in the most simple way. Examples include the Ising model, the Anderson model or the Hubbard model of condensed matter physics. 74 CHAPTER 2. SECOND QUANTIZATION

Figure 2.3: Dispersion of the normal modes, Eq. (2.52), of the 1d chain with periodic boundary conditions.

qj+N (t)= qj(t) for all j [for solutions for the case of a ﬁnite system, see Prob- lem 5]. We start with looking for particular (real) solutions of the following form13 i(−ωt+jl) qj(t)= e + c.c., (2.50) which, inserted into the equation of motion, yield for any j

mω2 q + q∗ = k eil 2+ e−il q + q∗ , (2.51) − j j − j j resulting in the following relation between ω and k(dispersion relation)14:

l k ω2(l)= ω2 sin2 , ω2 =4 . (2.52) 0 2 0 m

Here ω0 is just the eigenfrequency of a spring with constant k, and the prefactor 2 arises from the fact that each particle interacts with two neighbors. While 0 the condition (2.52) is independent of the amplitudes qj , i.e. of the initial conditions, we still need to account for the boundary (periodicity) condition. Inserting it into the solution (2.50) gives the following condition for l, indepen- n N dently of ω: l ln = N 2π, where n = 0, 1, 2, 2 . Thus there exists a discrete spectrum→ of N frequencies of modes± ± which···± can propagate along the chain (we have to exclude n = 0 since this corresponds to a time-independent trivial constant displacement),

k nπ N ω2 =4 sin2 , n = 1, 2, . (2.53) n m N ± ± ···± 2 This spectrum is shown in Fig. 2.3. These N solutions are the complete set of normal modes of the system (2.48), corresponding to its N degrees of freedom. These are collective modes in which all particles participate, all oscillate with the same frequency but with a well deﬁned phase which depends on the particle number. These normal modes are waves running along the chain with a phase velocity15 c ω /l . n ∼ n n 13 0 0 In principle, we could use a prefactor qj = q diﬀerent from one, but by rescaling of q it can always be eliminated. The key is that the amplitudes of all particles are strictly coupled. 14 2 x We use the relation 1 cos x = 2sin 2 . 15 − The actual phase velocity is ωn/kn, where the wave number kn = ln/a involves a length scale a which does not appear in the present discrete model. 2.4. INTERACTING PARTICLES 75

Due to the completeness of the system of normal modes, we can expand any excitation of particle j and the corresponding momentum pj(t) = mq˙j(t) into a supersposition of normal mode contributions (n = 0) 6

N N 2 2 1 n 1 0 i(−ωnt+2π N j) −iωnt qj(t) = Qn e = e Qn(j) (2.54) √N N √N N nX=− 2 nX=− 2 N N 2 2 1 n 1 0 i(−ωnt+2π N j) −iωnt pj(t) = Pn e = e Pn(j), (2.55) √N N √N N nX=− 2 nX=− 2 where P 0 = imω Q0 . Note that the complex conjugate contribution to mode n − n n n is contained in the sum (term n). Also, qj(t) and pj(t) are real functions. − ∗ By computing the complex conjugate qj and equating the result to qj we obtain 0 ∗ 0 the conditions (Qn) = Q−n and ω−n = ω−n. Analogously we obtain for the 0 ∗ 0 − momenta (Pn ) = P−n. To make the notation more compact we introduced the N-dimensional complex vectors Q~ n and P~n with the component j being 0 i2πnj/N 0 i2πnj/N 16 equal to Qn(j) = Qne and Pn(j) = Pn e . One readily proofs that these vectors form an orthogonal system by computing the scalar product (see problem 5)

N n+m ~ ~ 0 0 i2π N j 0 0 QnQm = QnQm e = NQnQmδn,−m. (2.56) j=1 X Using this property it is now straightforward to compute the hamilton function in normal mode representation. Consider ﬁrst the momentum contribution, N N N 1 2 2 p2(t)= P~ P~ e−i(ωn+ωm)t, (2.57) j N n m j=1 N N X nX=− 2 mX=− 2 where the sum over j has been absorbed in the scalar product. Using now the orthogonality condition (2.56) we immediately simplify

N p2(t)= P 0 2. (2.58) j | n | j=1 n X X 16See problem 5 76 CHAPTER 2. SECOND QUANTIZATION

Analogously, we compute the potential energy

N N k N k 2 2 U = [q (t) q (t)]2 = e−i(ωn+ωm)t 2 j − j+1 2N j=1 N N X nX=− 2 mX=− 2 N 0 0 i2π n j i2π n (j+1) i2π m j i2π m (j+1) Q Q e N e N e N e N . × n m − − j=1 X The sum over j can again be simpliﬁed using the orthogonality condition (2.56), which allows to replace m by n, − N 1 0 0 i2π n j i2π n (j+1) i2π m j i2π m (j+1) Q Q e N e N e N e N = N n m − − j=1 X i2π n i2π m = 1 e N 1 e N Q~ Q~ = − − n m 2πn ω2 = 2 1 cos δ Q0 Q0 =4 n δ Q0 2, − N n,−m n m ω2 n,−m| n| 0 2 x where we have used Eq. (2.52) and the relation 1 cos x = 2sin 2 . This yields for the potential energy −

k mω2 U = n 2 k n X and for the total hamilton function

N 2 1 m H(P,Q)= P 0 2 + ω2 Q0 2 . (2.59) 2m| n | 2 n| n| n=− N X2 Problem 5: Proof the orthogonality relation (2.56).

2.4.2 Quantization of the 1d chain We now quantize the interacting system (2.48) by replacing coordinates and momenta of all particles by operators

(q ,p ) (ˆq , pˆ ) , i =1,...N, i i → i i † † ~ withq î =q ˆj, pî =p î, [ˆqi, pˆj]= i δij. (2.60) 2.4. INTERACTING PARTICLES 77

The Hamilton function (2.48) now becomes an operator of the same functional form (correspondence principle),

N pˆ2 k Hˆ (ˆp, qˆ)= j + (ˆq qˆ )2 , 2m 2 j − j+1 j=1 X and we still use the periodic boundary conditionsq ˆN+i =q ˆi. The normal modes of the classical system remain normal modes in the quantum case as 0 0 well, only the amplitudes Qn and Pn become operators

N 2 1 −iωnt qˆj(t) = e Qˆn(j) (2.61) √N N nX=− 2 N 2 1 −iωnt pj(t) = e Pˆn(j), (2.62) √N N nX=− 2 ˆ ˆ0 ˆ ˆ0 ˆ0 where Qn(j) = Qn exp i2πnj/N , Pn(j) = Pn exp i2πnj/N and Pn = ˆ0 { } { } imωnQn. − ˆ0 What remains is to impose the necessary restrictions on the operators Qn ˆ0 and Pn such that they guarantee the properties (2.60). One readily verifies ˆ0 † ˆ0 ˆ0 † ˆ0 that hermiticity of the operators is fulfilled if (Q )n = Q−n, (P )n = P−n and ω−n = ωn. Next, consider the commutator ofq î andp ˆj and use the normal mode representations− (2.61, 2.62),

1 0 0 −i(ω +ω )t i 2π (kn+jm) [ˆq , pˆ ]= [Qˆ , Pˆ ]e n m e N . (2.63) k j N n m n m X X ~ ˆ0 ˆ0 A suﬃcient condition for this expression to be equal i δk,j is evidently [Qn, Pm]= i~δn,−m which is veriﬁed separately for the cases k = j and k = j. In other words, the normal mode operators obey the commutation relation6

ˆ0 ˆ0 † ~ Qn, (Pm) = i δn,m, (2.64) h i and the hamiltonian becomes, in normal mode representation,

N 2 1 m Hˆ (P,ˆ Qˆ)= Pˆ0 2 + ω2 Qˆ0 2 . (2.65) 2m| n | 2 n| n| n=− N X2 This is a superposition of N independent linear harmonic oscillators with the frequencies ωn given by Eq. (2.53). Applying the results for the superposition 78 CHAPTER 2. SECOND QUANTIZATION of oscillators, Sec. 2.3.2, we readily can perform the second quantization by mωn N N deﬁning dimensionless coordinates, ξn = ~ Qn, n = 2 ,... 2 , n = 0, and introducing the creation and annihilation operators, − 6 p 1 ∂ an = + ξn , (2.66) √2 ∂ξn † 1 ∂ † an = + ξn , [an, ak]= δn,k. (2.67) √2 −∂ξn Thus the hamiltonian and its eigenfunctions and eigenvalues can be written as

N 2 1 Hˆ = ~ω a† a + n n n 2 n=− N X2 N N a 0 = 0, n = ,... n| i − 2 2 1 m1 mN ψ = m ...m = a† ... a† 0 m1,...mN 1 N − N N | i √m1! ...mN ! 2 2 | i N 2 1 E = ~ω m + . n n 2 n=− N X2 Here 0 0 ... 0 = 0 ... 0 [N factors] denotes the ground state and a | i≡| i | i | i general state m−N/2 ...mN/2 = m−N/2 ... mN/2 contains mn elementary | i | i | i † excitations of the normal mode n, created by mn times applying operator an to the ground state.

Problem 6: The commutation relation (2.64) which was derived to satisfy the commutation relations of coordinates and momenta is that of bosons. This result was independent of whether the particles in the chain are fermions or bosons. Discuss this seeming contradiction.

2.4.3 Generalization to arbitrary interaction Of course, the simple 1d chain is a model with a limited range of applicability. A real system of N interacting particles in 1d will be more difficult, at least by three issues: first, the pair interaction potential V may have any form. Second, the interaction, in general, involves not only nearest neighbors, and third, the effect of the full 3d geometry may be relevant. We, therefore, now return to 2.4. INTERACTING PARTICLES 79 the general 3d system of N classical particles (2.1) with the total potential energy17 N U (q)= U(r )+ V (r r ), (2.68) tot i i − j i=1 1≤imass out for dimensional reasons)

λ mQ = Q , n =1,... 3N. (2.71) n n H n Since is real, symmetric and positive deﬁnite20 the eigenvalues are real H and positive corresponding to the normal mode frequencies ωn = √λn. Fur- thermore, as a result of the diagonalization, the 3N-dimensional eigenvectors

17Here we follow the discussion of Ref. [HKL+09] 18 (0) Recall that q, qs and ξ are 3N-dimensional vectors in conﬁguration space. 19Strictly speaking, from the 3N degrees of freedom, up to three [depending on the symmetry of U] may correspond to rotations of the whole system (around one of the three coordinate axes, these are center of mass excitations which do not change the particle distance), and the remaining are oscillations. 20 (0) qs corresponds to a mininmum, so the local curvature of Utot is positive in all directions 80 CHAPTER 2. SECOND QUANTIZATION

form a complete orthogonal system Qn with the scalar product QnQm 3N { } ≡ i=1 Qn(i)Qm(i) δm,n which means that any excitation can be expanded into a superposition∼ of the eigenvectors (normal modes), P 3N (0) q(t)= qs + cn(t)Qn. (2.72) n=1 X The expansion coeﬃcients cn(t) (scalar functions) are the normal coordinates. Their equation of motion is readily obtained by inserting a Taylor expansion of the gradient of Utot [analogous to (2.70)] into (2.69), ∂U 0= mq¨ + tot = mq¨ + ξ, (2.73) ∂q H· and, using Eq. (2.72) forq ¨ and eliminating with the help of (2.71), H 3N 2 0= m c¨n(t)+ cn(t)ωn Qn. (2.74) n=1 X Due to the orthogonality of the Qn which are non-zero, the solution of this equation implies that the terms in the parantheses vanish simultaneously for every n, leading to an equation for a harmonic oscillator with the solution c (t)= A cos ω t + B , n =1,... 3N, (2.75) n n { n n} where the coeﬃcients An and Bn depend on the initial conditions. Thus, the normal coordinates behave as independent linear 1d harmonic oscillators. In analogy to the coordinates, also the particle momenta, corresponding to some excitation q(t), can be expanded in terms of normal modes by using p(t) = mq˙(t). Using the result for cn(t), Eq. (2.75), we have the following general expansion

3N 3N q(t) q(0) = A cos ω t + B Q Q (t) (2.76) − s n { n n} n ≡ n n=1 n=1 X3N 3XN p(t) = A sin ω t + B P P (t), (2.77) n { n n} n ≡ n n=1 n=1 X X where the momentum amplitude vector is Pn = mωnQn. Finally, we can transform the Hamilton function into normal mode− representation, using the harmonic expansion (2.70) of the potential energy

p2 N p2 1 H(p,q)= + U (q)= U (0) + i + ξT (i) (s)ξ(j). (2.78) 2m tot s 2m 2 Hij i=1 X Xi6=j 2.4. INTERACTING PARTICLES 81

Eliminating the Hesse matrix with the help of (2.71) and inserting the expan- sions (2.76) and (2.77) we obtain

3N 3N ′ (0) Pn(t)Pn (t) m 2 H(p,q) U = + ω δ ′ Q (t)Q ′ (t) − s 2m 2 n n,n n n n=1 ′ X nX=1 3N P 2(t) m = n + ω2Q2 (t) H(P,Q), (2.79) 2m 2 n n ≡ n=1 X where, in the last line, the orthogonality of the eigenvectors has been used. Thus we have succeeded to diagonalize the hamiltonian of the N-particle system with arbitrary interaction. Assuming weak excitations from a stationary state the hamiltonian can be written as a superposition of 3N normal modes. This means, we can again apply the results from the case of uncoupled harmonic oscillators, Sec. 2.3.2, and immediately perform the “ﬁrst” and “second” quantization.

2.4.4 Quantization of the N-particle system For the ﬁrst quantization we have to replace the normal mode coordinates and momenta by operators, Q (t) Qˆ (t)= A cos ω t + B Qˆ n → n n { n n} n P (t) Pˆ (t)= A sin ω t + B Pˆ , (2.80) n → n n { n n} n leaving the time-dependence of the classical system unchanged. Further we have to make sure that the standard commutation relations are fulﬁlled, i.e. [Qˆn, Pˆm] = i~δn,m. This should follow from the commutation relations of the original particle coordinates and momenta, [xiα,pjβ] = i~δi,jδα,β, where α, β =1, 2, 3 and i, j =1,...N, see Problem 7. Then, the Hamilton operator becomes, in normal mode representation

3N Pˆ2(t) m Hˆ (P,ˆ Qˆ) = n + ω2Qˆ2 (t) , (2.81) 2m 2 n n n=1 ( ) X which allows us to directly introduce the creation and annihilation operators mωn ˆ by introducing ξn = ~ Qn, n =1,... 3N) p 1 ∂ an = + ξn , (2.82) √2 ∂ξn † 1 ∂ † an = + ξn , [an, ak]= δn,k. (2.83) √2 −∂ξn 82 CHAPTER 2. SECOND QUANTIZATION

Thus the hamiltonian and its eigenfunctions and eigenvalues can be written as

3N 1 Hˆ = ~ω a† a + n n n 2 n=1 X a 0 = 0, n =1,... 3N n| i 1 † n1 † n3N ψn1,...n3N = n1 ...n3N = (a1) ... (a3N ) 0 | i √n1! ...n3N ! | i 3N 1 E = ~ω n + . n n 2 n=1 X

Here 0 0 ... 0 = 0 ... 0 [3N factors] denotes the ground state and a | i≡| i | i | i general state n1 ...n3N = n1 ... n3N contains nn elementary excitations of | i | i | i † the normal mode n, created by nn times applying operator an to the ground state. In summary, in ﬁnding the normal modes of the interacting N-particle system the description is reduced to a superposition of independent contributions from 3N degrees of freedom. Depending on the system dimensionality, these include (for a three-dimsional system) 3 translations of the center of mass and 3 rotations of the system as a whole around the coordinate axes. The remaining normal modes correspond to excitations where the particle distances change. Due to the stability of the stationary state with respect to weak excitations, these relative excitations are harmonic oscillations which have been quantized. In other words, we have 3N 6 phonon modes associated with the corresponding creation and annihilation− operators and energy quanta. The frequencies of the modes are determined by the local curvature of the total potential energy (the diagonal elements of the Hesse matrix).

Problem 7: Proof the commutation relation [Qˆn, Pˆm]= i~δn,m. Problem 8: Apply the concept of the eigenvalue problem of the Hesse matrix to the solution of the normal modes of the 1d chain. Rederive the normal mode representation of the hamiltonian and check if the time dependencies vanish. 2.5. CONTINUOUS SYSTEMS 83 2.5 Continuous systems

2.5.1 Continuum limit of 1d chain So far we have considered discrete systems containing N point particles. If the number of particles grows and their spacing becomes small we will eventually reach a continuous system – the 1d chain becomes a 1d string. We start with assigning particle i a coordinate xj = ja where j =0,...N, a is the constant interparticle distance and the total length of the system is l = Na, see Fig.

We again consider a macroscopic system which is now periodically repeated after length l, i.e. points x = 0 and x = l are identical21. In the discrete system we have an equally spaced distribution of masses m of point particles with a linear mass density ρ = m/a. The interaction between the masses is characterized by an elastic tension σ = κa where we relabeled the spring constant by κ. The continuum limit is now performed by simulataneously increasing the particle number and reducing a but requiring that the density and the tension remain unchanged,

a, m 0 −→ N,κ −→ ∞ l, ρ, σ = const.

We now consider the central quantity, the displacement of the individual particles qi(t) which now transforms into a continuous displacement ﬁeld q(x,t). Further, with the continuum limit, diﬀerences become derivatives and the sum over the particles is replaced by an integral according to

q (t) q(x,t) j −→ ∂q q q a j+1 − j −→ ∂x 1 l dx. −→ a j 0 X Z Instead of the Hamilton function (2.48) we now consider the Lagrange function which is the diﬀerence of kinetic and potential energy, L = T V , − 21Thus we have formally introduced N + 1 lattice points but only N are diﬀerent. 84 CHAPTER 2. SECOND QUANTIZATION

Figure 2.4: Illustration of the minimal action principle: the physical equation of motion corresponds to the tractory q(x,t) which minimizes the action, Eq. (2.86) at ﬁxed initial and ﬁnal points (ti, 0) and (tf ,l). which in the continuum limit transforms to

N m κ L(q, q˙) = (˙q )2 (q q )2 2 j − 2 j − j+1 j=1 X n o 1 l ∂q(x,t) 2 ∂q(x,t) 2 dx ρ σ (2.84) −→ 2 ∂t − ∂x Z0 ( ) The advantage of using the Lagrange function which now is a functional of the displacement ﬁeld, L = L[q(x,t)], is that there exists a very general method of ﬁnding the corresponding equations of motion – the minimal action principle.

2.5.2 Equation of motion of the 1d string We now define the one-dimensional Lagrange density L l L = dx [˙q(x,t),q′(x,t)], (2.85) L Z0 where Eq. (2.84) shows that Lagrange density of the spring depends only on two fields – the time derivativeq ˙ and space derivative q′ of the displacement field. The action is defined as the time integral of the Lagrange function between a fixed initial time ti and final time tf

tf tf l S = dtL = dt dx [˙q(x,t),q′(x,t)]. (2.86) L Zti Zti Z0 The equation of motion of the 1d string follows from minimizing the action with respect to the independent variables of [this “minimal action principle” will be discussed in detail in Chapter 1, Sec.L 1.1], for illustration, see Fig. 2.4,

tf l δ δ 0= δS = dt dx Lδq˙ + Lδq′ δq˙ δq′ Zti Z0 tf l = dt dx ρqδ˙ q˙ σq′ δq′ . (2.87) { − } Zti Z0 2.5. CONTINUOUS SYSTEMS 85

∂ ′ We now change the order of diﬀerentiation and variation, δq˙ = ∂t δq and δq = ∂ ∂x δq and perform partial integrations with respect to t in the ﬁrst term and x in the second term of (2.87)

tf l 0= dt dx ρq¨ σq′′ δq, (2.88) − { − } Zti Z0 where the boundary values vanish because one requires that the variation δq(x,t) are zero at the border of the integration region, δq(0,t)= δq(l,t) 0. Since this equation has to be fulfilled for any fluctuation δq(x,t) the term≡ in the parantheses has to vanish which yields the equation of motion of the 1d string ∂2q(x,t) ∂2q(x,t) σ κ c2 =0, with c = = a . (2.89) ∂t2 − ∂x2 ρ m r r This is a linear wave equation for the displacement field, and we introduced the phase velocity, i.e. the sound speed c. The solution of this equation can be written as i(kx−ωt) q(x,t)= q0e + c.c., (2.90) which, inserted into Eq. (2.89), yields the dispersion relation ω(k)= c k, (2.91) · i.e., the displacement of the string performs a wave motion with linear dispersion – we observe an acoustic wave where the wave number k is continuous. It is now interesting to compare this result with the behavior of the original discrete N particle system. There the oscillation frequencies ωn were given − 22 by Eq. (2.53), and the wave numbers are discrete kn = 2πn/Na with n = 1, N/2, and the maximum wave number is kmax = π/a. Obviously, the ±discrete···± system does not have a linear dispersion, but we may consider the small k limit and expand the sin to first order:

κ πn 2 c2 ak 2 ω2 4 =4 n = ck , (2.92) n ≈ m N a2 2 n i.e. for small k the discrete system has exactly the same dispersion as the continuous system. The comparison with the discrete system also gives a hint at the existence of an upper limit for the wave number in the continuous system. In fact, k cannot be larger than π/amin where amin is the minimal distance of neighboring particles in the “continuous medium”. The two dispersions are shown in Fig. 2.5.

22 The wave number follows from the mode numbers ln by dividing by a 86 CHAPTER 2. SECOND QUANTIZATION

Figure 2.5: Dispersion of the normal modes of the discrete 1D chain and of the associated continuous system – the 1D string. The dispersions agree for small k up to a kmax=π/a.

One may, of course, ask whether a continuum model has its own right of existence, without being a limit of a discrete system. In other words, this would correspond to a system with an infinite particle number and, correspondingly, an infinite number M of normal modes. While we have not yet discussed how to quantize continuum systems it is immediately clear that there should be problems if the number of modes is unlimited. In fact, the total energy contains a zero point contribution for each mode which, with M going to infinity, will diverge. This problem does not occur for any realistic system because the particle number is always finite (though, possibly large). But a pure continuum model will be only physically relevant if such divergencies are avoided. The solution is found by co-called “renormalization” procedures where a maximum k-value (a cut-off) is introduced. This maybe not easy to derive for any specific field theory, however, based on the information from discrete systems, such a cut-off can always be motivated by choosing a physically relevant particle number, as we have seen in this chapter.

Thus we have succeeded to perform the continuum limit of the 1d chain – the 1d string and derive and solve its equation of motion. The solution is a continuum of acoustic waves which are the normal modes of the medium which replace the discrete normal modes of the linear chain. Now the question remains how to perform a quantization of the continuous system, how to introduce creation and annihilation operators. To this end we have to develop a more general formalism which is called canonical quatization and which will be discussed in the next chapter. 2.6. SOLUTIONS OF PROBLEMS 87 2.6 Solutions of Problems

1. A simple equation for ψ0 is readily obtained by inserting the deﬁnition of a into Eq. (2.38),

′ 0= ψ0(ξ)+ ξψ0(ξ), (2.93)

−ξ2/2 with the solution ψ0(ξ) = C0e , where C0 follows from the normal- ∞ 2 1/2 −1/2 ization x0 −∞ dξψ0 = 1, with the result C0 = (π /x0) , where the phase is arbitrary and chosen to be zero. R 2. Proof: Using ψ a† = a ψ and Eq. (2.41), direct computation yields h | | i

† 1 n+1 † † n ψn+1 a ψn = ψ0 a a (a ) ψ0 . | | n!(n + 1)! | |

The ﬁnal result √n + 1 is obtainedp by induction, starting with n = 0.

3. This problem reduces to the previous one by applying hermitean conju- gation ∗ ψ a ψ = ψ a† ψ = √n h n−1| | ni n| | n−1

88 CHAPTER 2. SECOND QUANTIZATION Chapter 3

Fermions and bosons

We now turn to the quantum statistical description of many-particle systems. The indistinguishability of microparticles leads to a number of far-reaching consequences for the behavior of particle ensembles. Among them are the symmetry properties of the wave function. As we will see there exist only two diﬀerent symmetries leading to either Bose or Fermi-Dirac statistics. Consider a single nonrelativistic quantum particle described by the hamiltonian hˆ. The stationary eigenvalue problem is given by the Schr¨odinger equation hˆ φ = ǫ φ , i =1, 2,... (3.1) | ii i| ii where the eigenvalues of the hamiltonian are ordered, ǫ1 < ǫ2 < ǫ3 ... . The associated single-particle orbitals φi form a complete orthonormal set of states in the single-particle Hilbert space1

φi φj = δi,j, ∞ h | i φ φ = 1. (3.2) | iih i| i=1 X 3.1 Spin statistics theorem

We now consider the quantum mechanical state Ψ of N identical particles | i which is characterized by a set of N quantum numbers j1,j2,...,jN , meaning that particle i is in single-particle state φji . The states Ψ are elements of the N-particle Hilbert space which we deﬁne| asi the direct product| i of single-particle

1The eigenvalues are assumed non-degenerate. Also, the extension to the case of a continuous basis is straightforward.

89 90 CHAPTER 3. FERMIONS AND BOSONS

Figure 3.1: Example of the occupation of single-particle orbitals by 3 particles. Exchange of identical particles (right) cannot change the measurable physical properties, such as the occupation probability.

Hilbert spaces, N = 1 1 1 ... (N factors), and are eigenstates of the total hamiltonianH Hˆ , ⊗H ⊗H ⊗ Hˆ Ψ = E Ψ , j = j ,j ,... (3.3) | {j}i {j}| {j}i { } { 1 2 } The explicit structure of the N particle states is not important now and will be discussed later2. − Since the particles are assumed indistinguishable it is clear that all physical observables cannot depend upon which of the particles occupies which single particle state, as long as all occupied orbitals, i.e. the set j, remain unchainged. In other words, exchanging two particles k and l (exchanging their orbitals, jk jl) in the state Ψ may not change the probability density, cf. Fig. 3.1. The↔ mathematical formulation| i of this statement is based on the permutation operator Pkl with the action P Ψ = P Ψ = kl| {j}i kl| j1,...,jk,...jl,...,jN i = Ψ Ψ′ , k,l =1,...N, (3.4) | j1,...,jl,...jk,...,jN i≡| {j}i ∀ where we have to require Ψ′ Ψ′ = Ψ Ψ . (3.5) h {j}| {j}i h {j}| {j}i

Indistinguishability of particles requires PklHˆ = Hˆ and [Pkl, Hˆ ] = 0, i.e. Pkl and Hˆ have common eigenstates. This means Pkl obeys the eigenvalue problem P Ψ = λ Ψ = Ψ′ . (3.6) kl| {j}i kl| {j}i | {j}i 2In this section we assume that the particles do not interact with each other. The generalization to interacting particles will be discussed in Sec. 3.2.5. 3.2. N-PARTICLE WAVE FUNCTIONS 91

† Obviously, Pkl = Pkl, so the eigenvalue λkl is real. Then, from Eqs. (3.5) and (3.6) immediately follows

λ2 = λ2 =1, k,l =1,...N, (3.7) kl ∀ with the two possible solutions: λ = 1 and λ = 1. From Eq. (3.6) it follows that, for λ = 1, the wave function Ψ is symmetric− under particle exchange whereas, for λ = 1, it changes sign| (i.e.,i it is “anti-symmetric”). This result was− obtained for an arbitrary pair of particles and we may expect that it is straightforwardly extended to systems with more than two particles. Experience shows that, in nature, there exist only two classes of microparticles – one which has a totally symmetric wave function with respect to exchange of any particle pair whereas, for the other, the wave function is antisymmetric. The first case describes particles with Bose-Einstein statistics (“bosons”) and the second, particles obeying Fermi-Dirac statistics (“fermions”)3. The one-to-one correspondence of (anti-)symmetric states with bosons (fermions) is the content of the spin-statistics theorem. It was first proven by Fierz [Fie39] and Pauli [Pau40] within relativistic quantum field theory. Require- ments include 1.) Lorentz invariance and relativistic causality, 2.) positivity of the energies of all particles and 3.) positive definiteness of the norm of all states.

3.2 Symmetric and antisymmetric N-particle wave functions

We now explicitly construct the N-particle wave function of a system of many fermions or bosons. For two particles occupying the orbitals φj1 and φj2 , respectively, there are two possible wave functions: Ψ and| Ψi which| i | j1,j2 i | j2,j1 i follow from one another by applying the permutation operator P12. Since both wave functions represent the same physical state it is reasonable to eliminate this ambiguity by constructing a new wave function as a suitable linear combination of the two,

Ψ ± = C Ψ + A P Ψ , (3.8) | j1,j2 i 12 {| j1,j2 i 12 12| j1,j2 i} with an arbitrary complex coeﬃcient A12. Using the eigenvalue property of the permutation operator, Eq. (3.6), we require that this wave function has

3Fictitious systems with mixed statistics have been investigated by various authors, e.g. [MG64, MG65] and obey “parastatistics”. For a text book discussion, see Ref. [Sch], p. 6. 92 CHAPTER 3. FERMIONS AND BOSONS the proper symmetry,

P Ψ ± = Ψ ±. (3.9) 12| j1,j2 i ±| j1,j2 i The explicit form of the coeﬃcients in Eq. (3.8) is obtained by acting on ± this equation with the permutation operator and equating this to Ψj1,j2 , 2 ˆ ±| i according to Eq. (3.9), and using P12 = 1, P Ψ ± = C Ψ + A P 2 Ψ = 12| j1,j2 i 12 | j2,j1 i 12 12| j1,j2 i = C A Ψ Ψ , 12 {± 12| j2,j1 i±| j1,j2 i} which leads to the requirement A = λ, whereas normalization of Ψ ± 12 | j1,j2 i yields C12 =1/√2. The ﬁnal result is

± 1 ± Ψj1,j2 = Ψj1,j2 P12 Ψj1,j2 Λ Ψj1,j2 (3.10) | i √2 {| i± | i} ≡ 12| i where, ± 1 Λ = 1 P12 , (3.11) 12 √2{ ± } denotes the (anti-)symmetrization operator of two particles which is a linear combination of the identity operator and the pair permutation operator. The extension of this result to 3 fermions or bosons is straightforward. For 3 particles (1, 2, 3) there exist 6 = 3! permutations: three pair permutations, (2, 1, 3), (3, 2, 1), (1, 3, 2), that are obtained by acting with the permuation operators P12,P13,P23, respectively on the initial conﬁguration. Further, there are two permutations involving all three particles, i.e. (3, 1, 2), (2, 3, 1), which are obtained by applying the operators P13P12 and P23P12, respectively. Thus, the three-particle (anti-)symmetrization operator has the form

± 1 Λ = 1 P12 P13 P23 + P13P12 + P23P12 , (3.12) 123 √3!{ ± ± ± } where we took into account the necessary sign change in the case of fermions resulting for any pair permutation. This result is generalized to N particles where there exists a total of N! permutations, according to4

Ψ ± =Λ± Ψ , | {j}i 1...N | {j}i (3.13)

4This result applies only to fermions. For bosons the prefactor has to be corrected, cf. Eq. (3.25). 3.2. N-PARTICLE WAVE FUNCTIONS 93 with the deﬁnition of the (anti-)symmetrization operator of N particles,

1 Λ± = sign(P )Pˆ (3.14) 1...N √ N! PǫS XN where the sum is over all possible permutations Pˆ which are elements of the permutation group S . Each permutation P has the parity, sign(P )=( 1)Np , N ± which is equal to the number Np of successive pair permuations into which Pˆ can be decomposed (cf. the example N = 3 above). Below we will construct ± the (anti-)symmetric state Ψ{j} explicitly. But before this we consider an alternative and very eﬃcient| notationi which is based on the occupation number formalism. ± The properties of the (anti-)symmetrization operators Λ1...N are analyzed in Problem 1, see Sec. 3.8.

3.2.1 Occupation number representation

The original N-particle state Ψ{j} contained clear information about which particle occupies which state.| Of coursei this is unphysical, as it is in conﬂict with the indistinguishability of particles. With the construction of the sym- ± metric or anti-symmetric N-particle state Ψ{j} this information about the identity of particles is eliminated, an the only| informatioi n which remained is how many particles np occupy single-particle orbital φp . We thus may use a ± | i diﬀerent notation for the state Ψ{j} in terms of the occupation numbers np of the single-particle orbitals, | i

Ψ ± = n n ... n , n =0, 1, 2,..., p =1, 2,... (3.15) | {j}i | 1 2 i ≡ |{ }i p Here n denotes the total set of occupation numbers of all single-particle orbitals.{ } Since this is the complete information about the N-particle system these states form a complete system which is orthonormal by construction of the (anti-)symmetrization operators,

′ n n = δ ′ = δ ′ δ ′ ... h{ }|{ }i {n},{n } n1,n1 n2,n2 n n = 1. (3.16) |{ }ih{ }| X{n} The nice feature of this representation is that it is equally applicable to fermions and bosons. The only diﬀerence between the two lies in the possible values of the occupation numbers, as we will see in the next two sections. 94 CHAPTER 3. FERMIONS AND BOSONS

3.2.2 Fock space

In Sec. 3.1 we have introduced the N-particle Hilbert space N . In the following we will need either totally symmetric or totally anti-symmetricH states + − which form the sub-spaces N and N of the Hilbert space. Furthermore, below we will develop the formalismH H of second quantization by defining creation and annihilation operators acting on symmetric or anti-symmetric states. Obviously, the action of these operators will give rise to a state with N +1 or N 1 particles. Thus, we have to introduce, in addition, a more general space− containing states with different particle numbers: We define the symmetric (anti-symmetric) Fock space ± as the direct sum of symmetric (anti-symmetric) Hilbert spaces ± withF particle numbers N =0, 1, 2,... , HN + = + + ..., F H0 ∪H1 ∪H2 ∪ − = − − .... (3.17) F H0 ∪H1 ∪H2 ∪ Here, we included the vacuum state 0 = 0, 0,... which is the state without particles which belongs to both Fock| spaces.i | i

3.2.3 Many-fermion wave function Let us return to the case of two particles, Eq. (3.10), and consider the case j1 = j2. Due to the minus sign in front of P12, we immediately conclude that − Ψj1,j1 0. This state is not normalizable and thus cannot be physically realized.| i ≡ In other words, two fermions cannot occupy the same single-particle orbital – this is the Pauli principle which has far-reaching consequences for the behavior of fermions. We now construct the explicit form of the anti-symmetric wave function. This is particularly simple if the particles are non-interacting. Then, the total hamiltonian is additive5, N

Hˆ = hˆi, (3.18) i=1 X and all hamiltonians commute, [hˆi, hˆj] = 0, for all i and j. Then all particles have common eigenstates, and the total wave function (prior to anti- symmetrization) has the form of a product

Ψ = Ψ = φ (1) φ (2) ... φ (N) | {j}i | j1,j2,...jN i | j1 i| j2 i | jN i 5This is an example of an observable of single-particle type which will be discussed more in detail in Sec. 3.3.1. 3.2. N-PARTICLE WAVE FUNCTIONS 95 where the argument of the orbitals denotes the number (index) of the particle that occupies this orbital. As we have just seen, for fermions, all orbitals have to be diﬀerent. Now, the anti-symmetrization of this state can be performed − immediately, by applying the operator Λ1...N given by Eq. (3.14). For two particles, we obtain

− 1 Ψj1,j2 = φj1 (1) φj2 (2) φj1 (2) φj2 (1) = | i √2! {| i| i−| i| i} = 0, 0,..., 1,..., 1,... . (3.19) | i In the last line, we used the occupation number representation, which has everywhere zeroes (unoccupied orbitals) except for the two orbitals with numbers j1 and j2. Obviously, the combination of orbitals in the ﬁrst line can be written as a determinant which allows for a compact notation of the general wave function of N fermions as a Slater determinant, φ (1) φ (2) ... φ (N) | j1 i | j1 i | j1 i − 1 φj2 (1) φj2 (2) ... φj2 (N) Ψj1,j2,...j = | i | i | i = | N i √ ...... N! ......

= 0, 0,..., 1,..., 1,..., 1,..., 1,... . (3.20) | i In the last line, the 1’s are at the positions of the occupied orbitals. This becomes obvious if the system is in the ground state, then the N energetically lowest orbitals are occupied, j1 = 1,j2 = 2,...jN = N, and the state has the simple notation 1, 1,... 1, 0, 0 ... with N subsequent 1’s. Obviously, the anti-symmetric wave| function is normalizedi to one. As discussed in Sec. 3.2.1, the (anti-)symmetric states form an orthonormal basis in Fock space. For fermions, the restriction of the occupation numbers leads to a slight modiﬁcation of the completeness relation which we, therefore, repeat: ′ n n = δ ′ δ ′ ..., h{ }|{ }i n1,n1 n2,n2 1 1 ... n n = 1. (3.21) |{ }ih{ }| n1=0 n2=0 X X 3.2.4 Many-boson wave function The case of bosons is analyzed analogously. Considering again the two-particle case + 1 Ψj1,j2 = φj1 (1) φj2 (2) + φj1 (2) φj2 (1) = | i √2! {| i| i | i| i} = 0, 0,..., 1,..., 1,... , (3.22) | i 96 CHAPTER 3. FERMIONS AND BOSONS the main diﬀerence to the fermions is the plus sign. Thus, this wave function is not represented by a determinant, but this combination of products with positive sign is called a permanent. The plus sign in the wave function (3.22) has the immediate consequence that the situation j1 = j2 now leads to a physical state, i.e., for bosons, there is no restriction on the occupation numbers, except for their normalization

∞ n = N, n =0, 1, 2,...N, p. (3.23) p p ∀ p=1 X

Thus, the two single-particle orbitals φj1 and φj2 occuring in Eq. (3.22) can accomodate an arbitrary number of bosons.| i If,| fori example, the two particles are both in the state φ , the symmetric wave function becomes | ji Ψ + = 0, 0,..., 2,... = | j,ji | i 1 = C(nj) φj(1) φj(2) + φj(2) φj(1) , (3.24) √2! | i| i | i| i where the coeﬃcient C(nj) is introduced to assure the normalization condition + + Ψj,j Ψj,j = 1. Since the two terms in (3.24) are identical the normalization givesh 1| =i 4 C(n ) 2/2, i.e. we obtain C(n = 2) = 1/√2. Repeating this | j | j analysis for a state with an arbitrary occupation number nj, there will be nj! identical terms, and we obtain the general result C(nj)=1/√nj. Finally, if ∞ there are several states with occupation numbers n1,n2,... with p=1 np = N, −1/2 the normalization constant becomes C(n1,n2,...)=(n1! n2! ... ) . Thus, for + P the case of bosons action of the symmetrization operator Λ1...N , Eq. (3.14), on the state Ψj1,j2,...jN will not yield a normalized state. A normalized symmetric state is obtained| byi the following prescription,

+ 1 + Ψj1,j2,...jN = Λ1...N Ψj1,j2,...jN (3.25) | i √n1!n2!... | i

1 Λ+ = P.ˆ (3.26) 1...N √ N! PǫS XN −1/2 Hence the symmetric state contains the prefactor (N!n1!n2!...) in front of the permanent. An example of the wave function of N bosons is

k Ψ + = n n ...n , 0, 0,... , n = N, (3.27) | j1,j2,...jN i | 1 2 k i p p=1 X 3.2. N-PARTICLE WAVE FUNCTIONS 97

where np = 0, for all p k, whereas all orbitals with the number p>k are empty. In6 particular, the≤ energetically lowest state of N non-interacting bosons (ground state) is the state where all particles occupy the lowest orbital φ1 , + | i i.e. Ψj1,j2,...jN GS = N0 ... 0 . This eﬀect of a macroscopic population which is possible| onlyi for particles| i with Bose statistics is called Bose-Einstein con- densation. Note, however, that in the case of interaction between the particles, a permanent constructed from the free single-particle orbitals will not be an eigenstate of the system. In that case, in a Bose condensate a ﬁnite fraction of particles will occupy excited orbitals (“condensate depletion”). The construction of the N-particle state for interacting bosons and fermions is subject of the next section.

3.2.5 Interacting bosons and fermions So far we have assumed that there is no interaction between the particles and the total hamiltonian is a sum of single-particle hamiltonians. In the case of interactions, N

Hˆ = hˆi + Hˆint, (3.28) i=1 X and the N-particle wave function will, in general, deviate from a product of single-particle orbitals. Moreover, there is no reason why interacting particles should occupy single-particle orbitals φp of a non-interacting system. The solution to this problem is based| oni the fact that the (anti-)symmetric ± states, Ψ{j} = n , form a complete orthonormal set in the N-particle Hilbert| space,i cf.|{ Eq.}i (3.16). This means, any symmetric or antisymmetric state can be represented as a superposition of N-particle permanents or determinants, respectively,

Ψ ± = C± n (3.29) | {j}i {n}|{ }i {n},NX=const The eﬀect of the interaction between the particles on the ground state wave function is to “add” contributions from determinants (permanents) involving higher lying orbitals to the ideal wave function, i.e. the interacting ground state includes contributions from (non-interacting) excited states. For weak interaction, we may expect that energetically low-lying orbitals will give the dominating contribution to the wave function. For example, for two fermions, the dominating states in the expansion (3.29) will be 1, 1, 0,... , 1, 0, 1,... , 1, 0, 0, 1 ... , 0, 1, 1,, 0 ... and so on. The computation| of the groundi | state ofi an| interactingi | many-particlei system is thus transformed into the computation 98 CHAPTER 3. FERMIONS AND BOSONS

± of the expansion coeﬃcients C{n}. This is the basis of the exact diagonalization method or conﬁguration interaction (CI). It is obvious that, if we would have obtained the eigenfunctions of the interacting hamiltonian, it would be represented by a diagonal matrix in this basis whith the eigenvalues populating the diagonal.6

This approach of computing the N-particle state via a superposition of permanents or determinants can be extended beyond the ground state properties. Indeed, extensions to thermodynamic equilibrium (mixed ensemble where the superpositions carry weights proportional to Boltzmann factors, e.g. [SBF+11]) and also nonequilibrium versions of CI (time-dependent CI, TDCI) that use pure states are meanwhile well established. In the latter, ± the coeﬃcients become time-dependent, C{n}(t), whereas the orbitals remain ﬁxed. We will consider the extension of the occupation number formalism to the thermodynamic properties of interacting bosons and fermions in Chapter 4. Further, nonequilibrium many-particle systems will be considered in Chapter 7 where we will develop an alternative approach based on nonequilibrium Green functions.

The main problem of CI-type methods is the exponential scaling with the number of particles which dramatically limits the class of solvable problems. Therefore, in recent years a large variety of approximate methods has been developed. Here we mention multiconfiguration (MC) approaches such as MC Hartree or MC Hartree-Fock which exist also in time-dependent variants (MCTDH and MCTDHF), e.g. [MMC90] and are now frequently applied to interacting Bose and Fermi systems. In this method not only the coefficients C±(t) are optimized but also the orbitals are adapted in a time-dependent fashion. The main advantage is the reduction of the basis size, as sompared to CI. A recent time-dependent application to the photoionization of helium can be found in Ref. [HB11]. Another very general approach consists in subdivid- ing the N-particle state in various classes with different properties. This has been termed “Generalized Active Space” (or restricted active space) approach and is very promising due to its generality [HB12, HB13]. An overview on first results is given in Ref. [HHB14].

6This N-particle state can be constructed from interacting single-particle orbitals as well. These are called “natural orbitals” and are the eigenvalues of the reduced one-particle density matrix. For a discussion see [SvL13]. 3.3. SECOND QUANTIZATION FOR BOSONS 99 3.3 Second quantization for bosons

We have seen in Chapter 1 for the example of the harmonic oscillator that an elegant approach to quantum many-particle systems is given by the method of second quantization. Using properly defined creation and annihilation operators, the hamiltonian of various N-particle systems was diagonalized. The examples studied in Chapter 1 did not explicitly include an interaction contribution to the hamiltonian – a simplification which will now be dropped. We will now consider the full hamiltonian (3.28) and transform it into second quantization representation. While this hamiltonian will, in general, not be diagonal, nevertheless the use of creation and annihilation operators yields a quite efficient approach to the many-particle problem.

3.3.1 Creation and annihilation operators for bosons † We now introduce the creation operatora ˆi acting on states from the symmetric Fock space +, cf. Sec. 3.2.2. It has the property to increase the occupation F number ni of single-particle orbital φi by one. In analogy to the harmonic oscillator, Sec. 2.3 we use the following| i deﬁnition

aˆ† n n ...n ... = √n +1 n n ...n +1 ... (3.30) i | 1 2 i i i | 1 2 i i While in case of coupled harmonic oscillators this operator created an additional excitation in oscillator “i”, now its action leads to a state with an additional particle in orbital “i”. The associated annihilation operatora î of orbital φi is now constructed as the hermitean adjoint (we use this as its def- | i † † † inition) ofa î , i.e. [âi ] =âi, and its action can be deduced from the definition (3.30),

′ ′ aî n1n2 ...ni ... = n n aî n1n2 ...ni ... | i ′ |{ }ih{ }| | i X{n } ′ † ′ ′ ∗ = n n1n2 ...ni ... aî n1 ...ni ... = ′ |{ }ih | | i X{n } ′ i ′ = n +1 δ ′ δ ′ n = i {n},{n } ni,ni+1|{ }i {n′} X p = √n n n ...n 1 ... , (3.31) i | 1 2 i − i yielding the same explicit definition that is familiar from the harmonic os- 7 † cillator : the adjoint ofa î is indeed an annihilation operator reducing the 7See our results for coupled harmonic oscillators in section 2.3.2. 100 CHAPTER 3. FERMIONS AND BOSONS

occupation of orbital φi by one. In the third line of Eq. (3.31) we introduced the modiﬁed Kronecker| symboli in which the occupation number of orbital i is missing,

i ′ ′ ′ ′ δ{n},{n } = δn1,n1 ...δni−1,ni−1 δni+1,ni+1 .... (3.32) ik δ ′ = δ ′ ...δ ′ δ ′ ....δ ′ δ ′ .... (3.33) {n},{n } n1,n1 ni−1,ni−1 ni+1,ni+1 nk−1,nk−1 nk+1,nk+1

In the second line, this deﬁnition is extended to two missing orbitals. We now need to verify the proper bosonic commutation relations, which are given by the Theorem: The creation and annihilation operators deﬁned by Eqs. (3.30, 3.31) obey the relations

[â , aˆ ] = [â†, aˆ† ]=0, i, k, (3.34) i k i k ∀ † aî, aˆk = δi,k. (3.35) h i Proof of relation (3.35): Consider first the case i = k and evaluate the commutator acting on an arbitrary state 6

aˆ , aˆ† n =a ˆ √n +1 ...n ,...n +1 ... i k |{ }i i k | i k i h i aˆ† √n ...n 1,...n ... =0 − k i| i − k i Consider now the case i = k: Then

aˆ , aˆ† n = (n + 1) n n n = n , k k |{ }i k |{ }i − k|{ }i |{ }i h i which proves the statement since no restrictions with respect to i and k were made. Analogously one proves the relations (3.34), see problem 18. We now consider the second quantization representation of important operators.

Construction of the N-particle state As for the harmonic oscillator or any quantized ﬁeld, an arbitrary many- particle state can be constructed from the vacuum state by repeatedly applying

8From this property we may also conclude that the ladder operators of the harmonic oscillator have bosonic nature, i.e. the elementary excitations of the oscillator (oscillation quanta or phonons) are bosons. 3.3. SECOND QUANTIZATION FOR BOSONS 101 the creation operator(s). For example, single and two-particle states with the proper normalization are obtained via

1 =a ˆ† 0 , | i | i 0, 0 ... 1, 0,... =a ˆ† 0 , | i i | i 1 2 0, 0 ... 2, 0,... = aˆ† 0 , | i √2! i | i 0, 0 ... 1, 0,... 1, 0,... =a ˆ†aˆ† 0 , i = j, | i i j| i 6 where, in the second (third) line, the 1 (2) stands on position i, whereas in the last line the 1’s are at positions i and j. This is readily generalized to an arbitrary symmetric N-particle state according to9.

n1 n2 1 † † n1,n2,... = aˆ1 aˆ2 ... 0 (3.36) | i √n1!n2! ... | i

Particle number operators The operator † nî =âi aî (3.37) is the occupation number operator for orbital i because, for n 1, i ≥ aˆ†aˆ n =â†√n n ...n 1 ... = n n , i i|{ }i i i| 1 i − i i|{ }i whereas, for n = 0,a ˆ†aˆ n = 0. Thus, the symmetric state n is an i i i|{ }i |{ }i eigenstate ofn î with the eigenvalue coinciding with the occupation number ni of this state. In other words: alln î have common eigenfunctions with the hamiltonian and commute, [ˆni,H]=0. The total particle number operator is defined as

∞ ∞ ˆ † N = nî = aî aî, (3.38) i=1 i=1 X X because its action yields the total number of particles in the system: Nˆ n = ∞ ˆ |{ }i i=1 ni n = N n . Thus, also N commutes with the hamiltonian and has the|{ same}i eigenfunctions.|{ }i P 9The origin of the prefactors was discussed in Sec. 3.2.4 and is also analogous to the case of the harmonic oscillator Sec. 2.3. 102 CHAPTER 3. FERMIONS AND BOSONS

Single-particle operators Consider now a general single-particle operator10 deﬁned as

Bˆ1 = ˆbα, (3.39) α=1 X where ˆbα acts only on the variables associated with particle with number “α”. We will now transform this operator into second quantization representation. To this end we deﬁne the matrix element with respect to the single-particle orbitals b = i ˆb j , (3.40) ij h | | i and the generalized projection operator11

N Πˆ = i j , (3.41) ij | iαh |α α=1 X where i denotes the orbital i occupied by particle α. | iα Theorem: The second quantization representation of a single-particle operator is given by ∞ ∞ ˆ ˆ † B1 = bij Πij = bij aî aˆj (3.42) i,j=1 i,j=1 X X Proof: We first expand ˆb, for an arbitrary particle α into a basis of single-particle orbitals i = φ , | i | ii ∞ ∞ ˆb = i i ˆb j j = b i j , | ih | | ih | ij| ih | i,j=1 i,j=1 X X where we used the definition (3.40) of the matrix element. With this result we can transform the total operator (3.39), using the definition (3.41),

N ∞ ∞ Bˆ = b i j = b Πˆ , (3.43) 1 ij| iαh |α ij ij α=1 i,j=1 i,j=1 X X X 10Examples are the total momentum, total kinetic energy, angular momentum or potential energy of the system. 11For i = j this deﬁnition contains the standard projection operator on state i , i.e. i i , whereas for i = j this operator projects onto a transition, i.e. transfers an arbitrary| i particle| ih | from state j 6 to state i . | i | i 3.3. SECOND QUANTIZATION FOR BOSONS 103

We now express Πˆ ij in terms of creation and annihilation operators by an- alyzing its action on a symmetric state (3.25), taking into account that Πˆ ij + 12 commutes with the symmetrization operator Λ1...N , Eq. (3.26) ,

1 N Πˆ n = Λ+ i j j j ... j . (3.44) ij|{ }i √ 1...N | iαh |α ·| 1i| 2i | N i n1!n2! ... α=1 X The product state is constructed from all orbitals including the orbitals i and | i j . Among the N factors there are, in general, ni factors i and nj factors |ji. Let us consider two cases. | i |1),i j = i: Since the single-particle orbitals form an orthonormal basis, j j = 6 h | i 1, multiplication with j α reduces the number of occurences of orbital j in the product state by one, whereash | multiplication with i increases the number of | iα orbitals i by one. The occurence of nj such orbitals (occupied by nj particles) in the product state gives rise to an overall factor of nj because nj terms of the sum will yield a non-vanishing contribution. Finally, we compare this result to the properly symmetrized state which follows from n by increasing ni by one and decreasing nj by one will be denoted by |{ }i

n i = n ,n ...n +1 ...n 1 ... { }j | 1 2 i j − i 1 + = Λ j1 j2 ... jN . (3.45) n ! ... (n + 1)! ... (n 1)! ... 1...N ·| i| i | i 1 i j − It contains thep same particle number N as the state n but, due to the diﬀer- +|{ }i ent orbital occupations, the prefactor in front of Λ1...N diﬀers by √nj/√ni + 1, compared to the one in Eq. (3.44) which we, therefore, can rewrite as

√ ˆ ni +1 i Πij n = nj n j |{ }i √nj { } † =a ˆ aˆj n . (3.46) i |{ }i

2), j = i: The same derivation now leads again to a number nj of factors, whereas the square roots in Eq. (3.46) compensate, and we obtain

Πˆ n = n n jj|{ }i j |{ }i =a ˆ†aˆ n . (3.47) j j|{ }i 12 From the deﬁnition (3.41) it is obvious that Πˆ ij is totally symmetric in all particle indices. 104 CHAPTER 3. FERMIONS AND BOSONS

Thus, the results (3.46) and (3.47) can be combined to the operator identity

N i j =â†aˆ (3.48) | iαh |α i j α=1 X which, together with the definition (3.45), proves the theorem13. For the special case that the orbitals are eigenfunctions of an operator, ˆb φ = b φ —such as the single-particle hamiltonian, the corresponding α| ii i| ii matrix is diagonal, bij = biδij, and the representation (3.42) simplifies to

∞ ∞ ˆ † B1 = bi aî aî = bi nî, (3.49) i=1 i=1 X X where bi are the eigenvalues of ˆb. Equation (3.49) naturally generalizes the familiar spectral representation of quantum mechanical operators to the case of many-body systems with arbitrary variable particle number.

Two-particle operators A two-particle operator is of the form

1 N Bˆ = ˆb , (3.50) 2 2! α,β αX6=β=1 where ˆbα,β acts only on particles α and β. An example is the operator of the pair interaction, ˆbα,β w( rα rβ ). We introduce again matrix elements, now with respect to two-particle→ | states− | composed as products of single-particle orbitals, which belong to the two-particle Hilbert space = , H2 H1 ⊗H1 b = ij ˆb kl , (3.51) ijkl h | | i Theorem: The second quantization representation of a two-particle operator is given by 1 ∞ Bˆ = b aˆ†aˆ†aˆ aˆ (3.52) 2 2! ijkl i j l k i,j,k,lX=1 Proof: We expand ˆb for an arbitrary pair α, β into a basis of two-particle orbitals

13See problem 2. 3.3. SECOND QUANTIZATION FOR BOSONS 105 ij = φ φ , | i | ii| ji ∞ ∞ ˆb = ij ij ˆb kl kl = ij kl b , | ih | | ih | | ih | ijkl i,j,k,lX=1 i,j,k,lX=1 leading to the following result for the total two-particle operator,

1 ∞ N Bˆ = b i j k l . (3.53) 2 2! ijkl | iα| iβh |αh |β i,j,k,lX=1 αX6=β=1 The second sum is readily transformed, using the property (3.48) of the sigle- particle states. We ﬁrst extend the summation over the particles to include α = β,

N N N N i j k l = i k j l δ i l | iα| iβh |αh |β | iαh |α | iβh |β − k,j | iαh |α α=1 α=1 αX6=β=1 X Xβ=1 X =a ˆ†aˆ aˆ†aˆ δ aˆ†aˆ i k j l − k,j i l =a ˆ† aˆ†aˆ + δ aˆ δ aˆ†aˆ i j k k,j l − k,j i l † n† o =a ˆi aˆjaˆkaˆl. In the third line we have used the commutation relation (3.35). After exchanging the order of the two annihilators (they commute) and inserting this expression into Eq. (3.53), we obtain the ﬁnal result (3.52)14.

General many-particle operators The above results are directly extended to more general operators involving K particles out of N 1 N Bˆ = ˆb , (3.54) K K! α1,...αK α16=α2X6=...αK =1 and which have the second quantization representation

1 ∞ Bˆ = b aˆ† ... aˆ† aˆ ....aˆ (3.55) K K! j1...jkm1...mk j1 jk mk m1 j1...j m1...m =1 k X k 14Note that the order of the creation operators and of the annihilators, respectively, is arbitrary. In Eq. (3.52) we have chosen an ascending order of the creators (same order as the indices of the matrix element) and a descending order of the annihilators, since this leads to an expression which is the same for Bose and Fermi statistics, cf. Sec. 3.4.1. 106 CHAPTER 3. FERMIONS AND BOSONS where we used the general matrix elements with respect to k-particle product ˆ states, bj1...jkm1...mk = j1 ...jk b m1 ...mk . Note again the inverse ordering of the annihilation operators.h Obviously,| | thei result (3.55) includes the previous examples of single and two-particle operators as special cases.

3.4 Second quantization for fermions

We now turn to particles with half-integer spin, i.e. fermions, which are described by anti-symmetric wave functions and obey the Pauli principle, cf. Sec. 3.2.3.

3.4.1 Creation and annihilation operators for fermions As for bosons we wish to introduce creation and annihilation operators that should again allow to construct any many-body state out of the vacuum state according to [cf. Eq. (3.36)]

n1 n2 n ,n ,... =Λ− i . . . i = aˆ† aˆ† ... 0 . n =0, 1, (3.56) | 1 2 i 1...N | 1 N i 1 2 | i i Due to the Pauli principle we expect that there will be no additional prefactors resulting from multiple occupations of orbitals, as in the case of bosons15. So far we do not know how these operators look like explicitly. Their deﬁnition has to make sure that the N-particle states have the correct anti-symmetry and that application of any creator more than once will return zero. To solve this problem, consider the example of two fermions which can occupy the orbitals k or l. The two-particle state has the symmetry kl = lk , upon particle exchange. The anti-symmetrized state is constructed| i of the−| producti state of particle 1 in state k and particle 2 in state l and has the properties

Λ− kl =ˆa†aˆ† 0 = 11 = Λ− lk = aˆ† aˆ† 0 , (3.57) 1...N | i l k| i | i − 1...N | i − k l | i i.e., it changes sign upon exchange of the particles (third equality). This indicates that the state depends on the order in which the orbitals are ﬁlled, i.e., on the order of action of the two creation operators. One possible choice is used in the above equation and immediately implies that

aˆ† aˆ† +ˆa†aˆ† = [ˆa† , aˆ†] =0, k,l, (3.58) k l l k k l + ∀ 15The prefactors are always equal to unity because 1! = 1 3.4. SECOND QUANTIZATION FOR FERMIONS 107

Figure 3.2: Illustration of the phase factor α in the fermionic creation and annihiliation operators. The single-particle orbitals are assumed to be in a deﬁnite order (e.g. with respect to the energy eigenvalues). The position of the particle that is added to (removed from) orbital φk is characterized by the number αk of particles occupying orbitals to the left, i.e. with energies smaller than ǫk. where we have introduced the anti-commutator16. In the special case, k = l, 2 † we immediately obtain aˆk = 0, for an arbitrary state, in agreement with the Pauli principle. Calculating the hermitean adjoint of Eq. (3.58) we obtain that the anti-commutator of two annihilators vanishes as well,

[ˆa , aˆ ] =0, k,l. (3.59) k l + ∀

We expect that this property holds for any two orbitals k,l and for any N- particle state that involves these orbitals. Now we can introduce an explicit deﬁnition of the fermionic creation operator which has all these properties. The operator creating a fermion in orbital k of a general many-body state is deﬁned as17

aˆ† ...,n ,... = (1 n )( 1)αk ...,n +1,... , α = n (3.60) k| k i − k − | k i k l Xl

16This was introduced by P. Jordan and E. Wigner in 1927. 17 There can be other conventions which diﬀer from ours by the choice of the exponent αk which, however, is irrelevant for physical observables. 108 CHAPTER 3. FERMIONS AND BOSONS set of anti-symmetric states and using (3.60)

′ ′ aˆk ...,nk,... = n n aˆk ...,nk,... = | i ′ |{ }ih{ }| | i X{n } ′ † ′ ∗ = n n aˆk n ′ |{ }ih{ }| |{ }i X{n } ′ α′ k ′ = (1 n )( 1) k δ ′ δ ′ n k {n },{n} nk,nk+1 ′ − − |{ }i X{n } = (2 n )( 1)αk ...,n 1,... − k − | k − i n ( 1)αk ...,n 1,... ≡ k − | k − i ′ where, in the third line, we used deﬁnition (3.32). Also, αk = αk because the sum involves only occupation numbers that are not altered. Note that the factor 2 nk = 1, for nk = 1. However, for nk = 0 the present result is not correct, as− it should return zero. To this end, in the last line we have added the factor nk that takes care of this case. At the same time this factor does not alter the result for nk = 1. Thus, the factor 2 nk can be skipped entirely, and we obtain the expression for the fermionic annihilation− operator of a particle in orbital k

aˆ ...,n ,... = n ( 1)αk ...,n 1,... (3.61) k| k i k − | k − i Using the deﬁnitions (3.60) and (3.61) one readily proves the anti-commutation relations given by the Theorem: The creation and annihilation operators deﬁned by Eqs. (3.60) and (3.61) obey the relations

[â , aˆ ] = [â†, aˆ† ] =0, i, k, (3.62) i k + i k + ∀ † aî, aˆk = δi,k. (3.63) + h i Proof of relation (3.62): Consider, the case of two annihilators and the action on an arbitrary anti- symmetric state

[â , aˆ ] n = (â aˆ +â aˆ ) n , (3.64) i k +|{ }i i k k i |{ }i and consider first case i = k. Inserting the definition (3.61), we obtain

(ˆa )2 n n aˆ n ...n 1 ... =0, k |{ }i ∼ k k| 1 k − i 3.4. SECOND QUANTIZATION FOR FERMIONS 109 and thus the anti-commutator vanishes as well. Consider now the case18 i

The proof of relation (3.63) proceeds analogously and is subject of Problem 3, cf. Sec. 3.8. Thus we have proved all anti-commutation relations for the fermionic operators and confirmed that the definitions (3.60) and (3.61) obey all properties required for fermionic field operators. We can now proceed to use thes operators to bring arbitrary quantum-mechanical operators into second quantized form in terms of fermionic orbitals.

Particle number operators As in the case of bosons, the operator

† nî =âi aî (3.65) is the occupation number operator for orbital i because, for ni =0, 1, aˆ†aˆ n =â†( 1)αi n ...n 1 ... = n [1 (n 1)] n , i i|{ }i i − | 1 i − i i − i − |{ }i where the prefactor equals ni, for ni = 1 and zero otherwise. Thus, the anti- symmetric state n is an eigenstate ofn î with the eigenvalue coinciding with |{ }i 19 the occupation number ni of this state . The total particle number operator is defined as ∞ ∞ ˆ † N = nî = aî aî, (3.66) i=1 i=1 X X ˆ ∞ because its action yields the total particle number: N n = i=1 ni n = N n . |{ }i |{ }i |{ }i P 18This covers the general case of i = k, since i and k are arbitrary. 19This result, together with the anti-commutation6 relations for the operators a and a† proves the consistency of the definitions (3.60) and (3.61). 110 CHAPTER 3. FERMIONS AND BOSONS

Single-particle operators Consider now again a single-particle operator

Bˆ1 = ˆbα, (3.67) α=1 X and let us ﬁnd its second quantization representation. Theorem: The second quantization representation of a single-particle operator is given by ∞ ˆ † B1 = bij aˆi aˆj (3.68) i,j=1 X Proof: As for bosons, cf. Eq. (3.43), we have

N ∞ ∞ Bˆ = b i j = b Πˆ , (3.69) 1 ij| iαh |α ij ij α=1 i,j=1 i,j=1 X X X ˆ ˆ † where Πij was deﬁned by (3.41), and it remains to show that Πij =a ˆi aˆj, for fermions as well. To this end we consider action of Πˆ ij on an anti-symmetric state, taking into accont that Πˆ ij commutes with the anti-symmetrization op- − erator Λ1...N , Eq. (3.14),

1 N Πˆ n = sign(π) i j j j ... j . (3.70) ij|{ }i √ | iαh |α ·| 1iπ(1)| 2iπ(2) | N iπ(N) N! α=1 πǫS X XN If the product state does not contain the orbital j expression (3.70) vanishes, due to the orthogonality of the orbitals. Otherwise,| i let j = j. Then j j = 1, k h | ki and the orbital jk will be replaced by i , unless the state i is already present, then we again obtain| i zero due to the Pauli| i principle, i.e. | i

Πˆ n (1 n )n n i , (3.71) ij|{ }i ∼ − i j { }j where we used the notation (3.45). What remains is to ﬁgure out the sign change due to the removal of a particle from the i-th orbital and creation of one in the k-th orbital. To this end we ﬁrst “move” the (empty) orbital j | i past all orbitals to the left occupied by αj = p

20 lower then Ei leading to αi pair exchanges and sign changes . Taking into account the deﬁnitions (3.60) and (3.61) we obtain21

Πˆ n =( 1)αi+αj (1 n )n n i =ˆa†aˆ n (3.72) ij|{ }i − − i j { }j i j|{ }i which, together with the equation (3.69), proves the theorem. Thus, the second quantization representation of single-particle operators is the same for bosons and fermions.

Two-particle operators As for bosons, we now derive the second quantization representation of a two- particle operator Bˆ2. Theorem: The second quantization representation of a two-particle operator is given by 1 ∞ Bˆ = b aˆ†aˆ†aˆ aˆ (3.73) 2 2! ijkl i j l k i,j,k,lX=1 Proof: As for bosons, we expand Bˆ into a basis of two-particle orbitals ij = φ φ , | i | ii| ji 1 ∞ N Bˆ = b i j k l , (3.74) 2 2! ijkl | iα| iβh |αh |β i,j,k,lX=1 αX6=β=1 and transform the second sum

N N N N i j k l = i k j l δ i l | iα| iβh |αh |β | iαh |α | iβh |β − k,j | iαh |α α=1 α=1 αX6=β=1 X Xβ=1 X =a ˆ†aˆ aˆ†aˆ δ aˆ†aˆ i k j l − k,j i l =a ˆ† aˆ†aˆ + δ aˆ δ aˆ†aˆ i − j k k,j l − k,j i l = aˆn†aˆ†aˆ aˆ . o − i j k l 20 Note that, if i>j, the occupation numbers occuring in αi have changed by one compared to those in αj. 21One readily veriﬁes that this result applies also to the case j = i. Then the prefactor is just [1 (nj 1)]nj = nj, and αi = αj, resulting in a plus sign − − † Πˆ jj n = nj n =a ˆ aˆj n . |{ }i |{ }i j |{ }i 112 CHAPTER 3. FERMIONS AND BOSONS

In the third line we have used the anti-commutation relation (3.63). After exchanging the order of the two annihilators, which now leads to a sign change, and inserting this expression into Eq. (3.74), we obtain the ﬁnal result (3.73).

General many-particle operators

The above results are directly extended to a general K-particle operator, K N, which was deﬁned in Eq. (3.54). Its second quantization representation≤ is found to be

1 ∞ Bˆ = b aˆ† ... aˆ† aˆ ....aˆ (3.75) K K! j1...jkm1...mk j1 jk mk m1 j1...j m1...m =1 k X k where we used the general matrix elements with respect to k-particle product ˆ states, bj1...jkm1...mk = j1 ...jk b m1 ...mk . Note again the inverse ordering of the annihilation operatorsh which| | exactlyi agrees with the expression for a bosonic system. Obviously, the result (3.75) includes the previous examples of single and two-particle operators as special cases.

3.4.2 Matrix elements in Fock space We now further extend the analysis of the anti-symmetric Fock space. A con- venient orthonormal basis for a system of N fermions are the anti-symmetric states n , cf. Eq. (3.21). Then operators are completely deﬁned by their action|{ on}i these states and by their matrix elements. For fermions the occupation number representation can be cast into a simple spinor formulation which we consider next.

Spinor representation of single-particle states

The fact that the fermionic occupation numbers have only two possible values is very similar to the two spin projections of spin 1/2 particles and allows for a very intuitive description in terms of spinors. Thus, an empty or singly occupied orbital can be written as a column

1 0 empty state, (3.76) | i → 0 − 0 1 occupied state, (3.77) | i → 1 − 3.4. SECOND QUANTIZATION FOR FERMIONS 113 and, analogously for the “bra”-states, 0 1 0 empty state, (3.78) h | → − 1 0 1 occupied state, (3.79) h | → − where the two form an orthonormal basis with 0 0 = 1 1 = 1 and 0 1 = 0. h | i h | i h | i Spinor representation of operators In the spinor representation each second quantization operator becomes a 2 2 matrix, × A A Aˆ 00 01 , (3.80) → A10 A11 where Aαβ = α Aˆ β and α, β =0, 1. The particleh | number| i operator has the following action 1 nˆ = 0, (3.81) 0 0 0 nˆ = 1 , (3.82) 1 1 and is, therefore, given by a diagonal matrix in this spinor representation with its eigenvalues on the diagonal,22 0 0 nˆ n nˆ n′ = n′1=ˆ , (3.83) →h | | i 0 1 and one readily conﬁrms that this is consistent with the action of the operator given by Eqs. (3.81) and (3.82).

Spinor representation of aˆ and aˆ† Using the deﬁnitions (3.60) and (3.61) we readily obtain the matrix elements of the creation and annihilation operator. We again consider the matrices with respect to single-particle states φ and take into account that, for fermions, | ki nk is either 0 or 1. As a result, we obtain

† ′ 0 0 † nk aˆ n = = δn ,1δn ,n′ +1 , (3.84) k k 1 0 k k k ≡Ak D E ′ 0 1 nk aˆk n = = δn ,0δn ,n′ −1 k, (3.85) h | | ki 0 0 k k k ≡A 22The ﬁrst [second] row corresponds to the case n = 0 [ n = 1 ], whereas the ﬁrst [second] column corresponds to n = 0 [ n = 1 ].h | h | h | h | | i | i | i | i 114 CHAPTER 3. FERMIONS AND BOSONS

† where the matrix ofa ˆk is the transposed of that ofa ˆk and we introduced the short-hand notation for the matrix δ δ ′ in the space of single-particle k nk,0 nk,1 orbitals φ 23. A | ki

Matrix elements of aˆ and aˆ† in Fock space

It is now easy to extend this to matrix elements with respect to anti-symmetric N-particle states. These matrices will have the same structure as (3.84) and (3.85), with respect to orbital k, and be diagonal with respect to all other orbitals. In addition, there will be a sign factor depending on the position of orbital k in the N-particle state, cf. deﬁnitions (3.60) and (3.61),

† ′ αk k † n aˆ n =( 1) δ ′ (3.86) { } k { } − {n},{n }Ak D E where the original prefactor 1 n′ has been transformed into an additional − k Kronecker delta for nk. The matrix of the annihilation operator is

′ αk k n aˆ n =( 1) δ ′ (3.87) h{ }| k| { }i − {n},{n } Ak

Matrix elements of one-particle operators in Fock space

To compute the matrix elements of one-particle operators, Eq. (3.43), we need the matrix of the projector Πˆ kl. Using the results (3.87) for the annihiliator

23 † We summarize the main properties of the matrices k and k which are a consequence † A A of the properties ofa ˆk anda ˆk and can be veriﬁed by direct matrix multiplication:

2 1. 2 = † = 0. Ak Ak † 0 0 2. k = = nk1ˆk, – a diagonal matrix with the eigenvalues ofn ˆk on the AkA 0 1 diagonal, cf. Eq. (3.83).

† 1 0 † † 3. k = = (1 nk)1ˆk = 1 k, i.e. and k anti-commute. A Ak 0 0 − −AkA Ak A

† † † 4. For diﬀerent single-particle spaces, k = l, [ , l] =[ , ] =[ k, l] = 0. 6 Ak A + Ak Al + A A + 3.4. SECOND QUANTIZATION FOR FERMIONS 115 and (3.86) for the creator successively we obtain, for the case k = l, 6

n aˆ†aˆ n′ = n aˆ† n¯ n¯ aˆ n′ = { } l k { } { } l { } h{ }| k| { }i D E X{n¯} D E ′ α α¯l k l =( 1) k ( 1) δn¯ ,0δ ′ δ n¯ ,n′ −1δn ,1δ δn¯ +1,n − − k {n¯},{n } k k l {n¯},{n} l l {n¯} ′ X α kl α¯l =( 1) k δn ,1δ ′ ( 1) δn¯ ,0δn¯ ,n δn¯ ,n′ δn¯ ,n′ −1δn¯ +1,n − l {n},{n } − k k k l l k k l l n,¯ n¯ Xk α ′ kl † ′ =( 1) k l δ ′ , α ′ = n + n , (3.88) − {n},{n }Al Ak k l m m m

† ′ ′ n aˆ aˆ n = n nˆ n = n δ ′ . (3.89) { } k k { } h{ }| k| { }i k {n},{n } D E

With the results (3.88) and (3.89) we readily obtain the matrix representation of a single-particle operator, deﬁned by Eq. (3.67),

∞ n Bˆ n′ = b n aˆ†aˆ n′ (3.90) { } 1 { } lk { } l k { } D E k,l=1 D E X

diag First, for a diagonal operator B , blk = bkδkl, the result is simply

∞ N diag ′ n Bˆ n = δ ′ b n = δ ′ b . (3.91) { } 1 { } {n},{n } k k {n},{n } nk D E k=1 k=1 X X where, in the last equality, we have simpliﬁed the summation by including only the occupied orbitals which have the numbers n1,n2 ...nN .

24This result is contained in expression (3.88). Indeed, in the special case k = l we obtain ′ ′ kl k αk l m

For the general case of a non-diagonal operator it follows from (3.90)25

ˆ ′ ′ n B1 n = δ{n},{n } bnknk + { } { } ( D E Xk=1 N k+l+γkl nknl † ′ + ( 1) bnlnk δ{n},{n } nl nk , . (3.92) − A A ) kX6=l=1 where γkl = 1, for k

Matrix elements of two-particle operators in Fock space To compute the matrix elements of two-particle operators, Eq. (3.73), we need the matrix elements of four-operator products, which we transform, using the anti-commutation relations (3.63) according to aˆ†aˆ†aˆ aˆ = aˆ†aˆ aˆ†aˆ + δ aˆ†aˆ . (3.93) i j l k − i l j k jl i k Next, transform the matrix element of the ﬁrst term on the right,

n aˆ†aˆ aˆ†aˆ n′ = n aˆ†aˆ n¯ n¯ aˆ†aˆ n′ = { } i l j k { } { } i l { } { } j k { } D E {n¯} D ED E X αi¯l il α¯jk′ jk = ( 1) δ δn ,1δn¯ ,0δn ,0δn¯ ,1 ( 1) δ ′ δn¯ ,1δn′ ,0δn¯ ,0δn′ ,1, − {n},{n¯} i i l l × − {n¯},{n } j j k k X{n¯} ′ ′ where α¯jk = pnk. 26We ﬁrst rewrite il jk iljk δ δ ′ = δ ′ δn ,n δn ,n δ ′ δ ′ , {n},{n¯p} {n¯p},{n } {n},{n } j ¯j k ¯k n¯i,ni n¯l,nl n¯ n¯ n¯ n¯ {Xn¯p} i Xl j k Taking into account the other Kronecker deltas we can combine pairs and perform the remaining four summations, δ ′ δ = δ ′ and so on. n¯i n¯i,ni n¯i,0 ni,0 P 3.4. SECOND QUANTIZATION FOR FERMIONS 117

This is a general result which also contains the cases of equal index pairs. Then, proceeding as in footnote 24, we obtain the results for the special cases.

α ′ ′ jk † i=l: ( 1) j k δ ′ n − {n},{n } iAjAk

αil il † j=k: ( 1) δ ′ n − {n},{n } jAi Al

α ′ ik † l=j: ( 1) ik δ ′ (1 n ) − {n},{n } − l Ai Ak i=l, j=k: δ{n},{n′}ninj

α ′ lj † αil il † k=i: ( 1) lj δ ′ n +( 1) δ ′ δ − {n},{n } iAlAj − {n},{n } ijAi Al 3.4.3 Fock Matrix of the binary interaction Of particular importance is the occupation number matrix representation of the interaction potential. This is an example of a two-particle quantity the properties of which we discussed in section 3.4.2. But for this special case, we can make further progress27. Starting point is the pair interaction

1 N Vˆ = wˆ(α, β), (3.95) 2 αX6=β=1 with the second quantization representation (3.73)

1 Vˆ = w aˆ†aˆ†aˆ aˆ , (3.96) 2 ijkl i j l k Xijkl where the matrix elements are deﬁned as

3 3 ∗ ∗ wijkl = d xd yφi (x)φj (y)w(x, y)φk(x)φl(y), (3.97) Z and have the following symmetries

wijkl = = wjilk, (3.98) ∗ wijkl = wklij, (3.99) where property (3.99) follows from the symmetry of the potential w(x, y) = w(y, x). This allows us to eliminate double counting of pairs from the sum in

27M. Heimsoth contributed to this section. 118 CHAPTER 3. FERMIONS AND BOSONS

Eq. (3.96)28

∞ ∞ ˆ − † † V = wijklaˆjaˆi aˆkaˆl, (3.100) 1≤i

n Vˆ n′ m Vˆ m′ = h{ }| |{ }i → h{ }| |{ }i N N − † † ′ ′ ′ ′ ′ = wm m m m m aˆmj aˆmi aˆm aˆm m . (3.103) i j k l h{ }| k l |{ }i 1≤i

28We summarize the main steps: First, using the anti-commutation relations of the annihilators and perfoming an index change, we transform (the contribution k = l vanishes),

− wijklaˆlaˆk = (wijkl wijlk)ˆalaˆk = w aˆlaˆk. − ijkl kl k

† † † † (wijkl wijlk)ˆa aˆ aˆlaˆk = (wijkl wjikl wijlk + wjilk)ˆa aˆ aˆlaˆk = − i j − − i j ij,k

(3.61), and taking advantage of the operator order in (3.103)30, the operators can be evaluated, with the result

N N ′ i+j+k+l − ′ ˆ ′ ′ ′ ′ m V m = ( 1) wm m m m m mi,mj m m ,m , h{ }| |{ }i − i j k l h{ }| |{ }i k l 1≤i

N ′ − n Vˆ n = δ ′ w . (3.105) h{ }| |{ }i {n},{n } mimj mimj 1≤i

2. The two states are identical except for one orbital: the orbital mp with number p is present in state m but is missing in state m′ which, h{ }| |{ }i instead, contains an orbital mr with number r missing in m . Then the scalar product of the two N 2 particle states is nonzeroh{ only}| if both these states are annihilated and− Eq. (3.104) yields32

N−1 ′ ′ mpmr † p+r − n Vˆ n = δ ′ m′ ( 1) Θ(p, r, i) w ′ h{ }| |{ }i {n},{n }Amp A r − · mimpmimr 1≤i,i6=p,r X (3.106) ′ Here Θ(p, r, i)= 1, if either mp

3. The two states are identical except for two orbitals with the numbers mp ′ ′ ′ and mq in m and mr and ms in m , respectively. Without loss of h{ }| |{ ′ }i ′ generality we can use mp

3.5 Coordinate representation of second quantization operators. Field operators

So far we have considered the creation and annihilation operators in an arbitrary basis of single-particle states. The coordinate and momentum representations are of particular importance and will be considered in the following. As before, an advantage of the present second quantization approach is that these considerations are entirely analogous for fermions and bosons and can be performed at once for both, the only diﬀerence being the details of the commutation (anticommutation) rules of the respective creation and annihilation operators. Here we start with the coordinate representation whereas the momentum representation will be introduced below, in Sec. 3.6.

3.5.1 Definition of field operators We now introduce operators that create or annihilate a particle at a given space point rather than in given orbital φi(r). To this end we consider the superposition in terms of the functions φi(r) where the coefficients are the creation and annihilation operators, ∞

ψˆ(x) = φi(x)ˆai, (3.108) i=1 X∞ ˆ† ∗ † ψ (x) = φi (x)ˆai . (3.109) i=1 X