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 hamil- tonian hˆ. The stationary eigenvalue problem is given by the Schr¨odinger equa- tion 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 con- tinuous basis is straightforward.
87 88 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 89
† 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 re- spect to exchange of any particle pair whereas, for the other, the wave func- tion 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 (fer- mions) 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 com- bination 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. 90 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 op- erators 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 91 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. 92 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 fol- lowing 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 cre- ation 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 gen- eral 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 par- ticles 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 93 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 num- bers 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 94 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 95
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 construc- tion 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 deter- minants, 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 96 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 repre- sented 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 prop- erties. 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 97 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 op- erators, the hamiltonian of various N-particle systems was diagonalized. The examples studied in Chapter 1 did not explicitly include an interaction con- tribution 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 ad- ditional excitation in oscillator “i”, now its action leads to a state with an additional particle in orbital “i”. The associated annihilation operatora ˆi of orbital φi is now constructed as the hermitean adjoint (we use this as its def- | i † † † inition) ofa ˆi , i.e. [ˆai ] =ˆai, and its action can be deduced from the definition (3.30),
′ ′ aˆi n1n2 ...ni ... = n n aˆi n1n2 ...ni ... | i ′ |{ }ih{ }| | i X{n } ′ † ′ ′ ∗ = n n1n2 ...ni ... aˆi 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 ˆi is indeed an annihilation operator reducing the 7See our results for coupled harmonic oscillators in section 2.3.2. 98 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
[ˆa , aˆ ] = [ˆa†, aˆ† ]=0, i, k, (3.34) i k i k ∀ † aˆi, 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 arbi- trary 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 99 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 † nˆi =ˆai aˆi (3.37) is the occupation number operator for orbital i because, for n 1, i ≥ aˆ†aˆ n =ˆa†√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 ˆi with the eigenvalue coinciding with the occupation number ni of this state. In other words: alln ˆi have common eigenfunctions with the hamiltonian and commute, [ˆni,H]=0. The total particle number operator is defined as
∞ ∞ ˆ † N = nˆi = aˆi aˆi, (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. 100 CHAPTER 3. FERMIONS AND BOSONS
Single-particle operators Consider now a general single-particle operator10 defined as
N
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 opera- tor is given by ∞ ∞ ˆ ˆ † B1 = bij Πij = bij aˆi 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 101
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. 102 CHAPTER 3. FERMIONS AND BOSONS
Thus, the results (3.46) and (3.47) can be combined to the operator identity
N i j =ˆa†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 aˆi aˆi = bi nˆi, (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 103 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 ex- changing 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. 104 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 de- scribed 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 105
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 oper- ator 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. 106 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 [ˆa , aˆ ] = [ˆa†, aˆ† ] =0, i, k, (3.62) i k + i k + ∀ † aˆi, 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 [ˆa , aˆ ] n = (ˆa aˆ +ˆa 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 107 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 op- erators 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 oper- ators 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 † nˆi =ˆai aˆi (3.65) is the occupation number operator for orbital i because, for ni =0, 1, aˆ†aˆ n =ˆa†( 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 ˆi with the eigenvalue coinciding with |{ }i 19 the occupation number ni of this state . The total particle number operator is defined as ∞ ∞ ˆ † N = nˆi = aˆi aˆi, (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). 108 CHAPTER 3. FERMIONS AND BOSONS Single-particle operators Consider now again a single-particle operator N Bˆ1 = ˆbα, (3.67) α=1 X and let us find its second quantization representation. Theorem: The second quantization representation of a single-particle opera- tor 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 110 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 occupa- tion 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 111 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 112 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 113 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 represen- tation 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 N 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 = p 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. 116 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 anni- hilators 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 quan- tization operators. Field operators So far we have considered the creation and annihilation operators in an arbi- trary basis of single-particle states. The coordinate and momentum represen- tations 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 annihila- tion 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 3.5. FIELD OPERATORS 119 Figure 3.3: Illustration of the relation of the field operators to the second quantization operators defined on a general basis φi(x) . The field operator ψˆ†(x) creates a particle at space point x (in spin state{ σ}) to which all single- | i particle orbitals φi contribute. The orbitals are vertically shifted for clarity. Here x = (r,σ), i.e. φi(x) is an eigenstate of the operator ˆr, and the φi(x) form a complete orthonormal set. Obviously, these operators have the desired property to create (annihilate) a particle at space point r in spin state σ. From the (anti-)symmetrization properties of the operators a and a† we immediately obtain ψˆ(x), ψˆ(x′) = 0, (3.110) ∓ h i ψˆ†(x), ψˆ†(x′) = 0, (3.111) ∓ h i ψˆ(x), ψˆ†(x′) = δ(x x′). (3.112) ∓ − h i These relations are straightforwardly proven by direct insertion of the def- initions (3.108) and (3.109). We demonstrate this for the last expression. ∞ ˆ ˆ† ′ ∗ ′ † ψ(x), ψ (x ) = φi(x)φj (x ) aˆi, aˆj = ∓ ∓ h i i,j=1 h i X∞ ∗ ′ ′ ′ = φ (x)φ (x )= δ(x x )= δ(r r )δ ′ , i i − − σ,σ i=1 X where, in the last line, we used the representation of the delta function in terms of a complete set of functions. 120 CHAPTER 3. FERMIONS AND BOSONS 3.5.2 Representation of operators We now transform operators into second quantization representation using the field operators, taking advantage of the identical expressions for bosons and fermions. Single-particle operators The general second-quantization representation was given by [cf. Secs. 3.3, 3.4] ∞ Bˆ = i ˆb j a†a . (3.113) 1 h | | i i j i,j=1 X We now transform the matrix element to coordinate representation: i ˆb j = dxdx′φ∗(x) x ˆb x′ φ (x′), (3.114) h | | i i h | | i j Z and obtain for the operator, taking into account the definitions (3.108) and (3.109), ∞ Bˆ = dxdx′ a†φ∗(x) x ˆb x′ φ (x′)a = 1 i i h | | i j j i,j=1 X Z = dxdx′ ψˆ†(x) x ˆb x′ ψˆ(x′). (3.115) h | | i Z For the important case that the matrix is diagonal, x ˆb x′ = ˆb(x)δ(x x′), the final expression simplifies to h | | i − † Bˆ1 = dx ψˆ (x)ˆb(x)ψˆ(x) (3.116) Z Consider a few important examples. The first is again the density operator. In first quantization the operator of the particle density for N particles follows from quantizing the classical result for point particles, N nˆ(x)= δ(x x ), (3.117) − α α=1 X 3.5. FIELD OPERATORS 121 33 and the expectation value in a certain N-particle state Ψ(x1,x2,...xN ) is N nˆ (x) = Ψ δ(x x ) Ψ h i h | − α | i α=1 X = N d2d3 ...dN Ψ(1, 2,...N) 2 = n(r,σ), (3.118) | | Z which is the single-particle spin density of a (in general correlated) N-particle system. The second quantization representation of the density operator follows from our above result (3.116) by replacing ˆb δ(x x′), i.e. → − nˆ(x) = dx′ψˆ†(x′)δ(x x′)ψˆ(x′)= ψˆ†(x)ψˆ(x), (3.119) − Z and the operator of the total density is the sum (integral) over all coordinate- spin states Nˆ = dx nˆ(x)= dx ψˆ†(x)ψˆ(x), (3.120) Z Z naturally extending the previous results for a discrete basis to continuous states. The second example is the kinetic energy operator which is also diagonal and has the second-quantized representation 1 Tˆ = dx ψˆ†(x) 2 ψˆ(x). (3.121) −2∇ Z The third example is the second quantization representation of the single- particle potential v(r) given by Vˆ = dx ψˆ†(x)v(r)ψˆ(x). (3.122) Z Two-particle operators In similar manner we obtain the field operator representation of a general two-particle operator 1 ∞ Bˆ = ij ˆb kl a†a†a a . (3.123) 2 2 h | | i i j l k i,j,k,lX=1 33This is the example of an (anti-)symmetrized pure state which is easily extended to mixed states. 122 CHAPTER 3. FERMIONS AND BOSONS We now transform the matrix element to coordinate representation: ij ˆb kl = dx dx dx dx φ∗(x )φ∗(x ) x x ˆb x x φ (x )φ (x ), (3.124) h | | i 1 2 3 4 i 1 j 2 h 1 2| | 3 4i l 3 k 4 Z and, assuming that the matrix is diagonal, x1x2 ˆb x3x4 = ˆb(x1,x2)δ(x1 x3)δ(x2 x4), we obtain, after inserting this resulth | into| (3.123),i − − 1 ∞ Bˆ = dx dx a†φ∗(x )a†φ∗(x )ˆb(x ,x )φ (x )a φ (x )a . 2 2 1 2 i i 1 j j 2 1 2 l 1 l k 2 k i,j,k,lX=1 Z Using again the defintion of the field operators the final result for a diagonal two-particle operator in coordinate representation is 1 Bˆ = dx dx ψˆ†(x )ψˆ†(x )ˆb(x ,x )ψˆ(x )ψˆ(x ) (3.125) 2 2 1 2 1 2 1 2 2 1 Z Note again the inverse ordering of the destruction operators which makes this result universally applicable to fermions and bosons. The most important example of this representation is the operator of the two-particle interaction, Wˆ , which is obtained by replacing ˆb(x ,x ) w(x x ). 1 2 → 1 − 2 3.6 Momentum representation of second quan- tization operators We now consider the momentum representation of the creation and annihila- tion operators. This is useful for translationally invariant systems such as the electron gas or the jellium model, since the eigenfunctions of the momentum operator, 1 ik·r x φk = φk (x)= e δ , x =(r,σ), (3.126) h | ,si ,s 1/2 s,σ are eigenfunctions of the translationV operator. Here we use periodic boundary conditions to represent an infinite system by a finite box of length L and volume = L3, so the wave numbers have discrete values k = 2πn /L,...k = V x x z 2πnz/L with nx,ny,nz being integer numbers. The eigenfunctions (3.126) form a complete orthonormal set, where the orthonormality condition reads 1 3 i(k′−k)r φk φk′ ′ = d re δ δ ′ = δk k′ δ ′ , (3.127) h ,s| ,s i s,σ s ,σ , s,s V σ V Z X where we took into account that the integral over the finite volume equals zero for k = k′ and otherwise. V 6 V 3.6. MOMENTUM REPRESENTATION 123 3.6.1 Creation and annihilation operators in momentum space The creation and annihilation operators on the Fock space of N-particle states constructed from the orbitals (3.126) are obtained by inverting the definition of the field operators (3.108) written with respect to the momentum-spin states (3.126) ψˆ(x)= φk′,σ′ (x)ak′,σ′ . k′ ′ Xσ ∗ Multiplication by φk,σ(x) and integrating over x yields, with the help of con- dition (3.127), ∗ 1 3 −ik·r ′ ′ ˆ ˆ ak,σ = dxφk σ ψ(x)= 1/2 d re ψ(r,σ), (3.128) Z V ZV and, similarly for the creation operator, † 1 3 ik·r ˆ† ak,σ = 1/2 d re ψ (r,σ). (3.129) V ZV Relations (3.128) and (3.129) are nothing but the Fourier transforms of the field operators. These operators obey the same (anti-)commutation relations as the field operators, which is a consequence of the linear superpositions (3.128), (3.129), cf. the proof of Eq. (3.112). 3.6.2 Representation of operators We again construct the second quantization representation of the relevant op- erators, now in terms of creation and annihilation operators in momentum space. Single-particle operators For a single-particle operator we have, according to our general result, Eq. (3.69), and denoting x =(r,s), x′ =(r′,s′), † ′ ′ Bˆ = a kσ ˆb k σ ak′ ′ 1 kσh | | i σ k k′ ′ Xσ Xσ ′ † ′ ′ ′ ′ = dxdx a kσ x x ˆb x x k σ ak′ ′ kσh | ih | | ih | i σ k k′ ′ Xσ Xσ Z 1 ′ † −ikr ′ ik′r′ = dxdx a e x ˆb x e ak′ ′ δ δ ′ ′ , (3.130) kσ h | | i σ σ,s σ ,s k k′ ′ V Xσ Xσ Z 124 CHAPTER 3. FERMIONS AND BOSONS where, in the last line, we inserted complete sets of momentum eigenstates (3.126). If the momentum matrix elements of the operator ˆb are known, the first line can be used directly. Otherwise, the matrix element is obtained from the the known coordinate space result in the last line. For an operator that commutes with the momentum operator and, thus, is given by a diagonal matrix one integration (and spin summation) can be performed. We demonstrate this for the example of the kinetic energy operator. 2 2 Then x ˆb x′ ~ ∇ δ(x x′), and we obtain, using the property (3.127), h | | i→− 2m − ~2 ′2 1 3 † −ikr k ik′r Tˆ = d r a e e a ′ kσ 2m k σ k k′ V V Xσ X Z ~2k2 = a† a . (3.131) 2m kσ kσ k Xσ In similar fashion we obtain for the single-particle potential, upon replacing x ˆb x′ v(r)δ(x x′), h | | i → − † 1 kr k′r ˆ ′ 3 −i i V = akσ ak σ d re v(r) e k k′ V Xσ X V Z † ′ ′ = v˜k−k akσ ak σ, (3.132) k k′ Xσ X where we introduced the Fourier transform of the single-particle potential, −1 3 −iqr v˜q = d rv(r)e . Finally, the operator of the single-particle density becomes,V in momentum space by Fourier transformation, R 1 3 † −iqr nˆq = nˆqσ = d rψσ(r)ψσ(r) e σ σ V X X V Z 1 † 1 3 i(k−k′)r −iqr = ak′σakσ d re e kk′ V V σX V Z 1 † = ak−q,σakσ. (3.133) k V Xσ This shows that the Fourier component of the density operator,n ˆq, describes a density fluctuation corresponding to a transition of a particle from state φkσ to state φk−q,σ , for arbitrary k. With this result we may rewrite the single-particle| i | potentiali (3.132) as Vˆ = v˜q nˆ−q. (3.134) V q X 3.6. MOMENTUM REPRESENTATION 125 Two-particle operators We now turn to two-particle operators. Rewriting the general result (3.73) for a spin-momentum basis, we obtain 1 † † ′ ′ ′ ′ Bˆ = a a k σ k σ ˆb k σ k σ ak′ ′ ak′ ′ (3.135) 2 2! k1σ1 k2σ2 h 1 1 2 2| | 1 1 2 2i 2σ2 1σ1 k k ′ ′ ′ ′ 1σX1 2σ2 k1σX1k2σ2 We now apply this result to the interaction potential where the matrix element ′ ′ ′ ′ in momentum representation was computed before, k1σ1k2σ2 wˆ k1σ1k2σ2 = ′ ′ ′ ′ ′ h | | i w˜(k1 k1)δk1+k2−k1−k2 δσ1,σ1 δσ2,σ2 ,andw ˜ denotes the Fourier transform of the pair interaction,− and the interaction does not change the spin of the involved particles, see problem 6, Sec. 3.8. Inserting this into Eq. (3.135) and intro- ducing the momentum transfer q = k′ k = k k′ , we obtain 1 − 1 2 − 2 1 Wˆ = w˜(q)a† a† a a , (3.136) 2! k1σ1 k2σ2 k2−q,σ2 k1+q,σ1 k k q σ1σ2 X1 2 X In similar manner other two-particle quantities are computed. With this result we can write down the second quantization representation in spin- momentum space of a generic hamiltonian that contains kinetic energy, an external potential and a pair interaction contribution. From the expressions (3.131, 3.132, 3.136) we obtain ~2 2 k † † Hˆ = a a + v˜ ′ a a ′ 2m kσ kσ k−k kσ k σ k kk′ Xσ Xσ 1 + w˜(q)a† a† a a . (3.137) 2! k1σ1 k2σ2 k2−q,σ2 k1+q,σ1 k k q σ1σ2 X1 2 X This result is a central starting point for many investigations in condensed matter physics, quantum plasmas or nuclear matter. 3.6.3 The uniform electron gas (jellium) An important special case where the momentum representation is advanta- geous is the uniform electron gas (UEG) or jellium. The system is spatially homogeneous, so the momentum is conserved. The hamiltonian of this system follows from the result (3.137) by omitting the external potential, ~2k2 ξ Hˆ = a† a + N M 2m kσ kσ 2 kσ X1 + w˜(q)a† a† a a . (3.138) 2! k1σ1 k2σ2 k2−q,σ2 k1+q,σ1 k k q σ1σ2 X1 2 X 126 CHAPTER 3. FERMIONS AND BOSONS The thermodynamic properties of the UEG will be discussed in Sec. 4.3.3. Application to relativistic quantum systems The momentum representation is conveniently extended to relativistic many- particle systems. In fact, since the Dirac equation of a free particle has plane wave solutions, we may use the same single-particle orbitals as in the non- relativistic case. With this, the matrix elements of the single-particle potential and of the interaction potential remain unchanged (if magnetic corrections to the interaction are neglected). The only change is in the kinetic energy con- tribution, due to the relativistic modification of the single-particle dispersion, 2 2 ǫ = ~ k √~2k2c2 + m2c4, where m is the rest mass. In the ultra-relativistic k 2m → limit, ǫk = ~ck. Otherwise the hamiltonian (3.137) remains valid. Of course, this is true only as long as pair creation processes are negligible. Otherwise we would need to extend the description by introducing the negative 2 2 2 2 4 energy branch ǫk− = √~ k c + m c and the corresponding second set of plane wave states. In− all matrix elements we would need to include a second index (+, ) referring to the energy band. − 3.7 Discussion and outlook After investigating the basic properties of the method of second quantization we now turn to more advanced topics. One of them is the extension of the analysis to systems at a finite temperature, i.e. in a mixed ensemble. This will be the subject of Chapter 4. After this we turn to the time evolution of the field operators following an external perturbation. This will be studied in detail for the case of single-time observables, in Chapter 5. A second route to nonequilibrium dynamics is to use field operator products that depend on two times which leads to the theory of nonequilibrium Green functions which we discuss in Chapter 7. 3.8 Problems to Chapter 3 ± ± 1. Express Λ123 via Λ12, cf. Eqs. (3.11) and (3.12). ± 2. Generalize the previous result to find a decomposition of Λ1...N into lower order operators. 3. Prove the bosonic commutation relations (3.34). 3.8. PROBLEMS TO CHAPTER 3 127 4. Prove the anti-commuation relation (3.63) between fermionic creation and annhiliation operators. 5. Discuss what happened to the sum over α in the derivation of Eq. (3.48). 6. Compute the momentum matrix element of the pair interaction.