Chapter 3 Fermions and bosons - Uni Kiel

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-reachingconsequences for the behavior of particle ensembles. Among them are thesymmetry properties of the wave function. As we will see there exist only twodifferent symmetries leading to either Bose or Fermi-Dirac statistics.

Consider a single nonrelativistic quantum particle described by the hamil-tonian h. The stationary eigenvalue problem is given by the Schrodinger equa-tion

h|φi〉 = ǫi|φi〉, i = 1, 2, . . . (3.1)

where the eigenvalues of the hamiltonian are ordered, ǫ1 < ǫ2 < ǫ3 . . . . Theassociated single-particle orbitals φi form a complete orthonormal set of statesin the single-particle Hilbert space1

〈φi|φj〉 = δi,j,∞∑

i=1

|φi〉〈φi| = 1. (3.2)

3.1 Spin statistics theorem

We now consider the quantum mechanical state |Ψ〉 of N identical particleswhich is characterized by a set of N quantum numbers j1, j2, ..., jN , meaningthat particle i is in single-particle state |φji〉. The states |Ψ〉 are elements of theN -particle Hilbert space which we define as the direct product 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 physicalproperties, such as the occupation probability.

Hilbert spaces, HN = H1 ⊗H1 ⊗H1 ⊗ . . . (N factors), and are eigenstates ofthe total hamiltonian H,

H|Ψ{j}〉 = E{j}|Ψ{j}〉, {j} = {j1, j2, . . . } (3.3)

The explicit structure of the N−particle states is not important now and willbe discussed later2.

Since the particles are assumed indistinguishable it is clear that all physicalobservables cannot depend upon which of the particles occupies which singleparticle 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 of this statement is based on the permutationoperator Pkl with the action

Pkl|Ψ{j}〉 = Pkl|Ψj1,...,jk,...jl,...,jN 〉 == |Ψj1,...,jl,...jk,...,jN 〉 ≡ |Ψ′

{j}〉, ∀k, l = 1, . . . N, (3.4)

where we have to require

〈Ψ′{j}|Ψ′

{j}〉 = 〈Ψ{j}|Ψ{j}〉. (3.5)

Indistinguishability of particles requires PklH = H and [Pkl, H] = 0, i.e. Pkl

and H have common eigenstates. This means Pkl obeys the eigenvalue problem

Pkl|Ψ{j}〉 = λkl|Ψ{j}〉 = |Ψ′{j}〉. (3.6)

2In this section we assume that the particles do not interact with each other. Thegeneralization to interacting particles will be discussed in Sec. 3.2.5.

3.2. N -PARTICLE WAVE FUNCTIONS 89

Obviously, P †kl = Pkl, so the eigenvalue λkl is real. Then, from Eqs. (3.5) and

(3.6) immediately follows

λ2kl = λ2 = 1, ∀k, l = 1, . . . N, (3.7)

with the two possible solutions: λ = 1 and λ = −1. From Eq. (3.6) it followsthat, for λ = 1, the wave function |Ψ〉 is symmetric under particle exchangewhereas, for λ = −1, it changes sign (i.e., it is “anti-symmetric”).

This result was obtained for an arbitrary pair of particles and we mayexpect that it is straightforwardly extended to systems with more than twoparticles. Experience shows that, in nature, there exist only two classes ofmicroparticles – 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-Einsteinstatistics (“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.) positivityof the energies of all particles and 3.) positive definiteness of the norm of allstates.

3.2 Symmetric and antisymmetric N-particle

wave functions

We now explicitly construct the N -particle wave function of a system of manyfermions or bosons. For two particles occupying the orbitals |φj1〉 and |φj2〉,respectively, there are two possible wave functions: |Ψj1,j2〉 and |Ψj2,j1〉 whichfollow from one another by applying the permutation operator P12. Since bothwave functions represent the same physical state it is reasonable to eliminatethis ambiguity by constructing a new wave function as a suitable linear com-bination of the two,

|Ψj1,j2〉± = C12 {|Ψj1,j2〉+ A12P12|Ψj1,j2〉} , (3.8)

with an arbitrary complex coefficient A12. Using the eigenvalue property ofthe 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.


the proper symmetry,

P12|Ψj1,j2〉± = ±|Ψj1,j2〉±. (3.9)

The explicit form of the coefficients in Eq. (3.8) is obtained by acting onthis equation with the permutation operator and equating this to ±|Ψj1,j2〉±,according to Eq. (3.9), and using P 2

12 = 1,

P12|Ψj1,j2〉± = C12

{

|Ψj2,j1〉+ A12P212|Ψj1,j2〉

}

=

= C12 {±A12|Ψj2,j1〉 ± |Ψj1,j2〉} ,

which leads to the requirement A12 = λ, whereas normalization of |Ψj1,j2〉±yields C12 = 1/

√2. The final result is

|Ψj1,j2〉± =1√2{|Ψj1,j2〉 ± P12|Ψj1,j2〉} ≡ Λ±

12|Ψj1,j2〉 (3.10)

where,

Λ±12 =

1√2{1± P12}, (3.11)

denotes the (anti-)symmetrization operator of two particles which is a linearcombination of the identity operator and the pair permutation operator.

The extension of this result to 3 fermions or bosons is straightforward. For3 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, thereare two permutations involving all three particles, i.e. (3, 1, 2), (2, 3, 1), whichare obtained by applying the operators P13P12 and P23P12, respectively. Thus,the three-particle (anti-)symmetrization operator has the form

Λ±123 =

1√3!{1± P12 ± P13 ± P23 + P13P12 + P23P12}, (3.12)

where we took into account the necessary sign change in the case of fermionsresulting for any pair permutation.

This result is generalized to N particles where there exists a total of N !permutations, according to4

|Ψ{j}〉± = Λ±1...N |Ψ{j}〉,

(3.13)

4This result applies only to fermions. For bosons the prefactor has to be corrected, cf.Eq. (3.25).


with the definition of the (anti-)symmetrization operator of N particles,

Λ±1...N =

1√N !

∑

PǫSN

sign(P )P (3.14)

where the sum is over all possible permutations P which are elements of thepermutation group SN . Each permutation P has the parity, sign(P ) = (±1)Np ,which is equal to the number Np of successive pair permuations into which Pcan be decomposed (cf. the example N = 3 above). Below we will constructthe (anti-)symmetric state |Ψ{j}〉± explicitly. But before this we consider analternative and very efficient notation which is based on the occupation numberformalism.

The properties of the (anti-)symmetrization operators Λ±1...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 whichparticle occupies which state. Of course this is unphysical, as it is in conflictwith the indistinguishability of particles. With the construction of the sym-metric or anti-symmetric N -particle state |Ψ{j}〉± this information about theidentity of particles is eliminated, an the only information which remained ishow many particles np occupy single-particle orbital |φp〉. We thus may use adifferent notation for the state |Ψ{j}〉± in terms of the occupation numbers np

of the single-particle orbitals,

|Ψ{j}〉± = |n1n2 . . . 〉 ≡ |{n}〉, np = 0, 1, 2, . . . , p = 1, 2, . . . (3.15)

Here {n} denotes the total set of occupation numbers of all single-particleorbitals. Since this is the complete information about the N -particle systemthese states form a complete system which is orthonormal by construction ofthe (anti-)symmetrization operators,

〈{n}|{n′}〉 = δ{n},{n′} = δn1,n′

1δn2,n′

2. . .

∑

{n}

|{n}〉〈{n}| = 1. (3.16)

The nice feature of this representation is that it is equally applicable tofermions and bosons. The only difference between the two lies in the possiblevalues of the occupation numbers, as we will see in the next two sections.


3.2.2 Fock space

In Sec. 3.1 we have introduced the N -particle Hilbert space HN . In the fol-lowing we will need either totally symmetric or totally anti-symmetric stateswhich form the sub-spaces H+

N and H−N of the Hilbert space. Furthermore,

below we will develop the formalism 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 + 1or N − 1 particles. Thus, we have to introduce, in addition, a more gen-eral space containing states with different particle numbers: We define thesymmetric (anti-symmetric) Fock space F± as the direct sum of symmetric(anti-symmetric) Hilbert spaces H±

N with particle numbers N = 0, 1, 2, . . . ,

F+ = H0 ∪H+1 ∪H+

2 ∪ . . . ,F− = H0 ∪H−

1 ∪H−2 ∪ . . . . (3.17)

Here, we included the vacuum state |0〉 = |0, 0, . . . 〉 which is the state withoutparticles which belongs to both Fock spaces.

3.2.3 Many-fermion wave function

Let us return to the case of two particles, Eq. (3.10), and consider the casej1 = 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 physicallyrealized. In other words, two fermions cannot occupy the same single-particleorbital – this is the Pauli principle which has far-reaching consequences for thebehavior 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 totalhamiltonian is additive5,

H =N∑

i=1

hi, (3.18)

and all hamiltonians commute, [hi, hj] = 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{j}〉 = |Ψj1,j2,...jN 〉 = |φj1(1)〉|φj2(2)〉 . . . |φjN (N)〉

5This is an example of an observable of single-particle type which will be discussed morein detail in Sec. 3.3.1.


where the argument of the orbitals denotes the number (index) of the particlethat occupies this orbital. As we have just seen, for fermions, all orbitals haveto be different. Now, the anti-symmetrization of this state can be performedimmediately, by applying the operator Λ−

1...N given by Eq. (3.14). For twoparticles, we obtain

|Ψj1,j2〉− =1√2!

{|φj1(1)〉|φj2(2)〉 − |φj1(2)〉|φj2(1)〉} =

= |0, 0, . . . , 1, . . . , 1, . . . 〉. (3.19)

In the last line, we used the occupation number representation, which haseverywhere zeroes (unoccupied orbitals) except for the two orbitals with num-bers j1 and j2. Obviously, the combination of orbitals in the first line can bewritten as a determinant which allows for a compact notation of the generalwave function of N fermions as a Slater determinant,

|Ψj1,j2,...jN 〉− =1√N !

∣

∣

∣

∣

∣

∣

∣

∣

|φj1(1)〉 |φj1(2)〉 ... |φj1(N)〉|φj2(1)〉 |φj2(2)〉 ... |φj2(N)〉... ... ... ...... ... ... ...

∣

∣

∣

∣

∣

∣

∣

∣

=

= |0, 0, . . . , 1, . . . , 1, . . . , 1, . . . , 1, . . . 〉. (3.20)

In the last line, the 1’s are at the positions of the occupied orbitals. Thisbecomes obvious if the system is in the ground state, then the N energeticallylowest orbitals are occupied, j1 = 1, j2 = 2, . . . jN = N , and the state hasthe simple notation |1, 1, . . . 1, 0, 0 . . . 〉 with N subsequent 1’s. Obviously, theanti-symmetric wave function is normalized to one.

As discussed in Sec. 3.2.1, the (anti-)symmetric states form an orthonormalbasis in Fock space. For fermions, the restriction of the occupation numbersleads to a slight modification of the completeness relation which we, therefore,repeat:

〈{n}|{n′}〉 = δn1,n′

1δn2,n′

2. . . ,

1∑

n1=0

1∑

n2=0

. . . |{n}〉〈{n}| = 1. (3.21)

3.2.4 Many-boson wave function

The case of bosons is analyzed analogously. Considering again the two-particlecase

|Ψj1,j2〉+ =1√2!

{|φj1(1)〉|φj2(2)〉+ |φj1(2)〉|φj2(1)〉} =

= |0, 0, . . . , 1, . . . , 1, . . . 〉, (3.22)


the main difference to the fermions is the plus sign. Thus, this wave functionis not represented by a determinant, but this combination of products withpositive sign is called a permanent.

The plus sign in the wave function (3.22) has the immediate consequencethat the situation j1 = j2 now leads to a physical state, i.e., for bosons, thereis no restriction on the occupation numbers, except for their normalization

∞∑

p=1

np = N, np = 0, 1, 2, . . . N, ∀p. (3.23)

Thus, the two single-particle orbitals |φj1〉 and |φj2〉 occuring in Eq. (3.22) canaccomodate an arbitrary number of bosons. If, for example, the two particlesare both in the state |φj〉, the symmetric wave function becomes

|Ψj,j〉+ = |0, 0, . . . , 2, . . . 〉 =

= C(nj)1√2!

{

|φj(1)〉|φj(2)〉+ |φj(2)〉|φj(1)〉}

, (3.24)

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 normalizationgives 1 = 4|C(nj)|2/2, i.e. we obtain C(nj = 2) = 1/

√2. Repeating this

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 ,

the normalization constant becomes C(n1, n2, ...) = (n1!n2! . . . )−1/2. Thus, for

the case of bosons action of the symmetrization operator Λ+1...N , Eq. (3.14), on

the state |Ψj1,j2,...jN 〉 will not yield a normalized state. A normalized symmetricstate is obtained by the following prescription,

|Ψj1,j2,...jN 〉+ =1√

n1!n2!...Λ+

1...N |Ψj1,j2,...jN 〉 (3.25)

Λ+1...N =

1√N !

∑

PǫSN

P . (3.26)

Hence the symmetric state contains the prefactor (N !n1!n2!...)−1/2 in front of

the permanent.An example of the wave function of N bosons is

|Ψj1,j2,...jN 〉+ = |n1n2 . . . nk, 0, 0, . . . 〉,k

∑

p=1

np = N, (3.27)


where np 6= 0, for all p ≤ k, whereas all orbitals with the number p > k areempty. In particular, the energetically lowest state ofN non-interacting bosons(ground state) is the state where all particles occupy the lowest orbital |φ1〉,i.e. |Ψj1,j2,...jN 〉+GS = |N0 . . . 0〉. This effect of a macroscopic population whichis possible only for particles 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 aneigenstate of the system. In that case, in a Bose condensate a finite fraction ofparticles will occupy excited orbitals (“condensate depletion”). The construc-tion of the N-particle state for interacting bosons and fermions is subject ofthe next section.

3.2.5 Interacting bosons and fermions

So far we have assumed that there is no interaction between the particles andthe total hamiltonian is a sum of single-particle hamiltonians. In the case ofinteractions,

H =N∑

i=1

hi + Hint, (3.28)

and the N -particle wave function will, in general, deviate from a product ofsingle-particle orbitals. Moreover, there is no reason why interacting particlesshould occupy single-particle orbitals |φp〉 of a non-interacting system.

The solution to this problem is based on the fact that the (anti-)symmetricstates, |Ψ{j}〉± = |{n}〉, form a complete orthonormal set in the N -particleHilbert space, cf. Eq. (3.16). This means, any symmetric or antisymmetricstate can be represented as a superposition of N -particle permanents or deter-minants, respectively,

|Ψ{j}〉± =∑

{n}, N=const

C±{n}|{n}〉 (3.29)

The effect of the interaction between the particles on the ground state wavefunction is to “add” contributions from determinants (permanents) involvinghigher lying orbitals to the ideal wave function, i.e. the interacting groundstate includes contributions from (non-interacting) excited states. For weakinteraction, we may expect that energetically low-lying orbitals will give thedominating 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 ground state ofan interacting many-particle system is thus transformed into the computation


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 haveobtained the eigenfunctions of the interacting hamiltonian, it would be repre-sented by a diagonal matrix in this basis whith the eigenvalues populating thediagonal.6

This approach of computing the N -particle state via a superposition ofpermanents or determinants can be extended beyond the ground state prop-erties. Indeed, extensions to thermodynamic equilibrium (mixed ensemblewhere 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 remainfixed. We will consider the extension of the occupation number formalism tothe thermodynamic properties of interacting bosons and fermions in Chapter 4.Further, nonequilibrium many-particle systems will be considered in Chapter 7where we will develop an alternative approach based on nonequilibrium Greenfunctions.

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

6This N -particle state can be constructed from interacting single-particle orbitals aswell. These are called “natural orbitals” and are the eigenvalues of the reduced one-particledensity 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 anelegant approach to quantum many-particle systems is given by the methodof second quantization. Using properly defined creation and annihilation op-erators, the hamiltonian of various N -particle systems was diagonalized. Theexamples 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 secondquantization representation. While this hamiltonian will, in general, not bediagonal, nevertheless the use of creation and annihilation operators yields aquite efficient approach to the many-particle problem.

3.3.1 Creation and annihilation operators for bosons

We now introduce the creation operator a†i acting on states from the symmetricFock space F+, cf. Sec. 3.2.2. It has the property to increase the occupationnumber ni of single-particle orbital |φi〉 by one. In analogy to the harmonicoscillator, Sec. 2.3 we use the following definition

a†i |n1n2 . . . ni . . . 〉 =√ni + 1 |n1n2 . . . ni + 1 . . . 〉 (3.30)

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 anadditional particle in orbital “i”. The associated annihilation operator ai oforbital |φi〉 is now constructed as the hermitean adjoint (we use this as its def-inition) of a†i , i.e. [a

†i ]† = ai, and its action can be deduced from the definition

(3.30),

ai|n1n2 . . . ni . . . 〉 =∑

{n′}

|{n′}〉〈{n′}|ai|n1n2 . . . ni . . . 〉

=∑

{n′}

|{n′}〉〈n1n2 . . . ni . . . |a†i |n′1 . . . n

′i . . . 〉∗ =

=∑

{n′}

√

n′i + 1 δi{n},{n′}δni,n′

i+1|{n′}〉 =

=√ni |n1n2 . . . ni − 1 . . . 〉, (3.31)

yielding the same explicit definition that is familiar from the harmonic os-cillator7: the adjoint of a†i is indeed an annihilation operator reducing the

7See our results for coupled harmonic oscillators in section 2.3.2.


occupation of orbital |φi〉 by one. In the third line of Eq. (3.31) we introducedthe modified Kronecker symbol in which the occupation number of orbital i ismissing,

δi{n},{n′} = δn1,n′

1. . . δni−1,n′

i−1δni+1,n′

i+1. . . . (3.32)

δik{n},{n′} = δn1,n′

1. . . δni−1,n′

i−1δni+1,n′

i+1. . . .δnk−1,n

′

k−1δnk+1,n

′

k+1. . . . (3.33)

In the second line, this definition is extended to two missing orbitals.

We now need to verify the proper bosonic commutation relations, whichare given by theTheorem: The creation and annihilation operators defined by Eqs. (3.30,3.31) obey the relations

[ai, ak] = [a†i , a†k] = 0, ∀i, k, (3.34)

[

ai, a†k

]

= δi,k. (3.35)

Proof of relation (3.35):Consider first the case i 6= k and evaluate the commutator acting on an arbi-trary state

[

ai, a†k

]

|{n}〉 = ai√nk + 1| . . . ni, . . . nk + 1 . . . 〉

− a†k√ni| . . . ni − 1, . . . nk . . . 〉 = 0

Consider now the case i = k: Then

[

ak, a†k

]

|{n}〉 = (nk + 1)|{n}〉 − nk|{n}〉 = |{n}〉,

which proves the statement since no restrictions with respect to i and k weremade. Analogously one proves the relations (3.34), see problem 18. We nowconsider 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 harmonicoscillator have bosonic nature, i.e. the elementary excitations of the oscillator (oscillationquanta or phonons) are bosons.


the creation operator(s). For example, single and two-particle states with theproper normalization are obtained via

|1〉 = a†|0〉,|0, 0 . . . 1, 0, . . . 〉 = a†i |0〉,

|0, 0 . . . 2, 0, . . . 〉 =1√2!

(

a†i

)2

|0〉,

|0, 0 . . . 1, 0, . . . 1, 0, . . . 〉 = a†i a†j|0〉, i 6= j,

where, in the second (third) line, the 1 (2) stands on position i, whereas inthe last line the 1’s are at positions i and j. This is readily generalized to anarbitrary symmetric N -particle state according to9.

|n1, n2, . . . 〉 =1√

n1!n2! . . .

(

a†1

)n1(

a†2

)n2

. . . |0〉 (3.36)

Particle number operators

The operator

ni = a†i ai (3.37)

is the occupation number operator for orbital i because, for ni ≥ 1,

a†i ai|{n}〉 = a†i√ni|n1 . . . ni − 1 . . . 〉 = ni|{n}〉,

whereas, for ni = 0, a†i ai|{n}〉 = 0. Thus, the symmetric state |{n}〉 is aneigenstate of ni with the eigenvalue coinciding with the occupation numberni of this state. In other words: all ni have common eigenfunctions with thehamiltonian and commute, [ni, H] = 0.

The total particle number operator is defined as

N =∞∑

i=1

ni =∞∑

i=1

a†i ai, (3.38)

because its action yields the total number of particles in the system: N |{n}〉 =∑∞

i=1 ni|{n}〉 = N |{n}〉. Thus, also N commutes with the hamiltonian andhas the same eigenfunctions.

9The origin of the prefactors was discussed in Sec. 3.2.4 and is also analogous to the caseof the harmonic oscillator Sec. 2.3.


Single-particle operators

Consider now a general single-particle operator10 defined as

B1 =N∑

α=1

bα, (3.39)

where bα 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-particleorbitals

bij = 〈i|b|j〉, (3.40)

and the generalized projection operator11

Πij =N∑

α=1

|i〉α〈j|α, (3.41)

where |i〉α denotes the orbital i occupied by particle α.

Theorem: The second quantization representation of a single-particle opera-tor is given by

B1 =∞∑

i,j=1

bij Πij =∞∑

i,j=1

bij a†i aj (3.42)

Proof:We first expand b, for an arbitrary particle α into a basis of single-particleorbitals |i〉 = |φi〉,

b =∞∑

i,j=1

|i〉〈i|b|j〉〈j| =∞∑

i,j=1

bij|i〉〈j|,

where we used the definition (3.40) of the matrix element. With this result wecan transform the total operator (3.39), using the definition (3.41),

B1 =N∑

α=1

∞∑

i,j=1

bij|i〉α〈j|α =∞∑

i,j=1

bijΠij, (3.43)

10Examples are the total momentum, total kinetic energy, angular momentum or potentialenergy of the system.

11For i = j this definition contains the standard projection operator on state |i〉, i.e. |i〉〈i|,whereas for i 6= j this operator projects onto a transition, i.e. transfers an arbitrary particlefrom state |j〉 to state |i〉.


We now express Πij in terms of creation and annihilation operators by an-

alyzing its action on a symmetric state (3.25), taking into account that Πij

commutes with the symmetrization operator Λ+1...N , Eq. (3.26)

12,

Πij|{n}〉 =1√

n1!n2! . . .Λ+

1...N

N∑

α=1

|i〉α〈j|α · |j1〉|j2〉 . . . |jN〉. (3.44)

The product state is constructed from all orbitals including the orbitals |i〉 and|j〉. Among the N factors there are, in general, ni factors |i〉 and nj factors|j〉. Let us consider two cases.1), j 6= i: Since the single-particle orbitals form an orthonormal basis, 〈j|j〉 =1, multiplication with 〈j|α reduces the number of occurences of orbital j in theproduct state by one, whereas multiplication with |i〉α increases the number oforbitals 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 ofthe sum will yield a non-vanishing contribution.

Finally, we compare this result to the properly symmetrized state whichfollows from |{n}〉 by increasing ni by one and decreasing nj by one will bedenoted by

∣

∣{n}ij⟩

= |n1, n2 . . . ni + 1 . . . nj − 1 . . . 〉

=1

√

n1! . . . (ni + 1)! . . . (nj − 1)! . . .Λ+

1...N · |j1〉|j2〉 . . . |jN〉. (3.45)

It contains the same particle number N as the state |{n}〉 but, due to the differ-ent orbital occupations, the prefactor in front of Λ+

1...N differs by√nj/

√ni + 1,

compared to the one in Eq. (3.44) which we, therefore, can rewrite as

Πij|{n}〉 = nj

√ni + 1√nj

∣

∣{n}ij⟩

= a†i aj|{n}〉. (3.46)

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

Πjj|{n}〉 = nj |{n}〉= a†j aj|{n}〉. (3.47)

12From the definition (3.41) it is obvious that Πij is totally symmetric in all particleindices.


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

N∑

α=1

|i〉α〈j|α = a†i aj (3.48)

which, together with the definition (3.45), proves the theorem13.For the special case that the orbitals are eigenfunctions of an operator,

bα|φi〉 = bi|φi〉—such as the single-particle hamiltonian, the correspondingmatrix is diagonal, bij = biδij, and the representation (3.42) simplifies to

B1 =∞∑

i=1

bi a†i ai =

∞∑

i=1

bi ni, (3.49)

where bi are the eigenvalues of b. Equation (3.49) naturally generalizes thefamiliar spectral representation of quantum mechanical operators to the caseof many-body systems with arbitrary variable particle number.

Two-particle operators

A two-particle operator is of the form

B2 =1

2!

N∑

α 6=β=1

bα,β, (3.50)

where bα,β acts only on particles α and β. An example is the operator of

the pair interaction, bα,β → w(|rα− rβ|). We introduce again matrix elements,now with respect to two-particle states composed as products of single-particleorbitals, which belong to the two-particle Hilbert space H2 = H1 ⊗H1,

bijkl = 〈ij|b|kl〉, (3.51)

Theorem: The second quantization representation of a two-particle operatoris given by

B2 =1

2!

∞∑

i,j,k,l=1

bijkl a†i a

†j alak (3.52)

Proof:We expand b for an arbitrary pair α, β into a basis of two-particle orbitals

13See problem 2.


|ij〉 = |φi〉|φj〉,

b =∞∑

i,j,k,l=1

|ij〉〈ij|b|kl〉〈kl| =∞∑

i,j,k,l=1

|ij〉〈kl| bijkl,

leading to the following result for the total two-particle operator,

B2 =1

2!

∞∑

i,j,k,l=1

bijkl

N∑

α 6=β=1

|i〉α|j〉β〈k|α〈l|β. (3.53)

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 includeα = β,

N∑

α 6=β=1

|i〉α|j〉β〈k|α〈l|β =N∑

α=1

|i〉α〈k|αN∑

β=1

|j〉β〈l|β − δk,j

N∑

α=1

|i〉α〈l|α

= a†i aka†j al − δk,j a

†i al

= a†i

{

a†j ak + δk,j

}

al − δk,j a†i al

= a†i a†j akal.

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 thisexpression 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 Kparticles out of N

BK =1

K!

N∑

α1 6=α2 6=...αK=1

bα1,...αK, (3.54)

and which have the second quantization representation

BK =1

K!

∞∑

j1...jkm1...mk=1

bj1...jkm1...mka†j1 . . . a

†jkamk

. . . .am1(3.55)

14Note that the order of the creation operators and of the annihilators, respectively, isarbitrary. In Eq. (3.52) we have chosen an ascending order of the creators (same order as theindices of the matrix element) and a descending order of the annihilators, since this leadsto an expression which is the same for Bose and Fermi statistics, cf. Sec. 3.4.1.


where we used the general matrix elements with respect to k-particle productstates, bj1...jkm1...mk

= 〈j1 . . . jk|b|m1 . . .mk〉. Note again the inverse ordering ofthe annihilation operators. Obviously, the result (3.55) includes the previousexamples 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 thatshould again allow to construct any many-body state out of the vacuum stateaccording to [cf. Eq. (3.36)]

|n1, n2, . . . 〉 = Λ−1...N |i1 . . . iN〉 =

(

a†1

)n1(

a†2

)n2

. . . |0〉. ni = 0, 1, (3.56)

Due to the Pauli principle we expect that there will be no additional prefactorsresulting from multiple occupations of orbitals, as in the case of bosons15. Sofar we do not know how these operators look like explicitly. Their definitionhas to make sure that the N -particle states have the correct anti-symmetryand that application of any creator more than once will return zero.

To solve this problem, consider the example of two fermions which canoccupy the orbitals k or l. The two-particle state has the symmetry |kl〉 =−|lk〉, upon particle exchange. The anti-symmetrized state is constructed ofthe product state of particle 1 in state k and particle 2 in state l and has theproperties

Λ−1...N |kl〉 = a†l a

†k|0〉 = |11〉 = −Λ−

1...N |lk〉 = −a†ka†l |0〉, (3.57)

i.e., it changes sign upon exchange of the particles (third equality). Thisindicates 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 choiceis used in the above equation and immediately implies that

a†ka†l + a†l a

†k = [a†k, a

†l ]+ = 0, ∀k, l, (3.58)

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 andannihiliation operators. The single-particle orbitals are assumed to be in adefinite order (e.g. with respect to the energy eigenvalues). The position ofthe particle that is added to (removed from) orbital φk is characterized by thenumber αk of particles occupying orbitals to the left, i.e. with energies smallerthan ǫk.

where we have introduced the anti-commutator16. In the special case, k = l,

we immediately obtain(

a†k

)2

= 0, for an arbitrary state, in agreement with

the Pauli principle. Calculating the hermitean adjoint of Eq. (3.58) we obtainthat the anti-commutator of two annihilators vanishes as well,

[ak, al]+ = 0, ∀k, l. (3.59)

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 orbitalk of a general many-body state is defined as17

a†k| . . . , nk, . . . 〉 = (1− nk)(−1)αk | . . . , nk + 1, . . . 〉, αk =∑

l<k

nl (3.60)

where the prefactor explicitly enforces the Pauli principle, and the sign factortakes into account the position of the orbital k in the many-fermion state andthe number of fermions standing “to the left” of the “newly created” particle,cf. Fig. 3.2. In other words, with αk pair exchanges (anti-commutations) theparticle would move from the leftmost place to the position (e.g. accordingto an ordering with respect to the orbital energies Ek) of orbital k in the N -particle state. We now derive the annihilation operator by inserting a complete

16This was introduced by P. Jordan and E. Wigner in 1927.17There can be other conventions which differ from ours by the choice of the exponent αk

which, however, is irrelevant for physical observables.


set of anti-symmetric states and using (3.60)

ak| . . . , nk, . . . 〉 =∑

{n′}

|{n′}〉〈{n′}|ak| . . . , nk, . . . 〉 =

=∑

{n′}

|{n′}〉〈{n}|a†k|{n′}〉∗

=∑

{n′}

(1− n′k)(−1)α

′

kδk{n′},{n}δnk,n′

k+1|{n′}〉

= (2− nk)(−1)αk | . . . , nk − 1, . . . 〉≡ nk(−1)αk | . . . , nk − 1, . . . 〉

where, in the third line, we used definition (3.32). Also, α′k = αk because the

sum involves only occupation numbers that are not altered. Note that thefactor 2 − nk = 1, for nk = 1. However, for nk = 0 the present result is notcorrect, as it should return zero. To this end, in the last line we have addedthe factor nk that takes care of this case. At the same time this factor does notalter the result for nk = 1. Thus, the factor 2−nk can be skipped entirely, andwe obtain the expression for the fermionic annihilation operator of a particlein orbital k

ak| . . . , nk, . . . 〉 = nk(−1)αk | . . . , nk − 1, . . . 〉 (3.61)

Using the definitions (3.60) and (3.61) one readily proves the anti-commutationrelations given by the

Theorem: The creation and annihilation operators defined by Eqs. (3.60) and(3.61) obey the relations

[ai, ak]+ = [a†i , a†k]+ = 0, ∀i, k, (3.62)

[

ai, a†k

]

+= δi,k. (3.63)

Proof of relation (3.62):Consider, the case of two annihilators and the action on an arbitrary anti-symmetric state

[ai, ak]+|{n}〉 = (aiak + akai) |{n}〉, (3.64)

and consider first case i = k. Inserting the definition (3.61), we obtain

(ak)2 |{n}〉 ∼ nkak|n1 . . . nk − 1 . . . 〉 = 0,


and thus the anti-commutator vanishes as well. Consider now the case18 i < k:

aiak|{n}〉 = aink(−1)∑

l<k nl |n1 . . . nk − 1 . . . 〉 == nink(−1)

∑l<k nl(−1)

∑l<i nl |n1 . . . ni − 1 . . . nk − 1 . . . 〉.

Now we compute the result of the action of the exchanged operator pair

akai|{n}〉 = akni(−1)∑

l<i nl |n1 . . . ni − 1 . . . 〉 == nink(−1)

∑l<i nl(−1)

∑l<k nl−1|n1 . . . ni − 1 . . . nk − 1 . . . 〉,

The only difference compared to the first result is in the additional −1 in thesecond exponent. It arises because, upon action of ak after ai, the number ofparticles to the left of k is already reduced by one. Thus, the two expressionsdiffer just by a minus sign, which proves vanishing of the anti-commutator.

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 propertiesrequired for fermionic field operators. We can now proceed to use thes oper-ators to bring arbitrary quantum-mechanical operators into second quantizedform in terms of fermionic orbitals.

Particle number operators

As in the case of bosons, the operator

ni = a†i ai (3.65)

is the occupation number operator for orbital i because, for ni = 0, 1,

a†i ai|{n}〉 = a†i (−1)αi |n1 . . . ni − 1 . . . 〉 = ni[1− (ni − 1)]|{n}〉,where the prefactor equals ni, for ni = 1 and zero otherwise. Thus, the anti-symmetric state |{n}〉 is an eigenstate of ni with the eigenvalue coinciding withthe occupation number ni of this state

19.The total particle number operator is defined as

N =∞∑

i=1

ni =∞∑

i=1

a†i ai, (3.66)

because its action yields the total particle number: N |{n}〉 = ∑∞i=1 ni|{n}〉 =

N |{n}〉.18This covers the general case of i 6= k, since i and k are arbitrary.19This result, together with the anti-commutation relations for the operators a and a†

proves the consistency of the definitions (3.60) and (3.61).



Consider now again a single-particle operator

B1 =N∑

α=1

bα, (3.67)

and let us find its second quantization representation.Theorem: The second quantization representation of a single-particle opera-tor is given by

B1 =∞∑

i,j=1

bij a†i aj (3.68)

Proof:As for bosons, cf. Eq. (3.43), we have

B1 =N∑

α=1

∞∑

i,j=1

bij|i〉α〈j|α =∞∑

i,j=1

bijΠij, (3.69)

where Πij was defined by (3.41), and it remains to show that Πij = a†i aj, for

fermions as well. To this end we consider action of Πij on an anti-symmetric

state, taking into accont that Πij commutes with the anti-symmetrization op-erator Λ−

1...N , Eq. (3.14),

Πij|{n}〉 =1√N !

N∑

α=1

∑

πǫSN

sign(π)|i〉α〈j|α · |j1〉π(1)|j2〉π(2) . . . |jN〉π(N). (3.70)

If the product state does not contain the orbital |j〉 expression (3.70) vanishes,due to the orthogonality of the orbitals. Otherwise, let jk = j. Then 〈j|jk〉 = 1,and the orbital |jk〉 will be replaced by |i〉, unless the state |i〉 is already present,then we again obtain zero due to the Pauli principle, i.e.

Πij|{n}〉 ∼ (1− ni)nj

∣

∣{n}ij⟩

, (3.71)

where we used the notation (3.45). What remains is to figure out the signchange due to the removal of a particle from the i-th orbital and creation ofone in the k-th orbital. To this end we first “move” the (empty) orbital |j〉past all orbitals to the left occupied by αj =

∑

p<j np particles, requiring justαj pair permutations and sign changes. Next we move the “new” particle toorbital “i” past αi =

∑

p<i np particles occupying the orbitals with an energy


lower then Ei leading to αi pair exchanges and sign changes20. Taking intoaccount the definitions (3.60) and (3.61) we obtain21

Πij|{n}〉 = (−1)αi+αj(1− ni)nj

∣

∣{n}ij⟩

= a†i aj|{n}〉 (3.72)

which, together with the equation (3.69), proves the theorem. Thus, the secondquantization representation of single-particle operators is the same for bosonsand fermions.


As for bosons, we now derive the second quantization representation of a two-particle operator B2.Theorem: The second quantization representation of a two-particle operatoris given by

B2 =1

2!

∞∑

i,j,k,l=1

bijkl a†i a

†j alak (3.73)

Proof:As for bosons, we expand B into a basis of two-particle orbitals |ij〉 = |φi〉|φj〉,

B2 =1

2!

∞∑

i,j,k,l=1

bijkl

N∑

α 6=β=1

|i〉α|j〉β〈k|α〈l|β, (3.74)

and transform the second sum

N∑

α 6=β=1

|i〉α|j〉β〈k|α〈l|β =N∑

α=1

|i〉α〈k|αN∑

β=1

|j〉β〈l|β − δk,j

N∑

α=1

|i〉α〈l|α

= a†i aka†j al − δk,j a

†i al

= a†i

{

−a†j ak + δk,j

}

al − δk,j a†i al

= −a†i a†j akal.20Note 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

Πjj |{n}〉 = nj |{n}〉 = a†j aj |{n}〉.


In the third line we have used the anti-commutation relation (3.63). Afterexchanging 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 isfound to be

BK =1

K!

∞∑

j1...jkm1...mk=1

bj1...jkm1...mka†j1 . . . a

†jkamk

. . . .am1(3.75)

where we used the general matrix elements with respect to k-particle productstates, bj1...jkm1...mk

= 〈j1 . . . jk|b|m1 . . .mk〉. Note again the inverse orderingof the annihilation operators which exactly agrees with the expression for abosonic system. Obviously, the result (3.75) includes the previous examples ofsingle 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-symmetricstates |{n}〉, cf. Eq. (3.21). Then operators are completely defined by theiraction on these states and by their matrix elements. For fermions the occupa-tion number representation can be cast into a simple spinor formulation whichwe consider next.

Spinor representation of single-particle states

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

|0〉 →(

10

)

− empty state, (3.76)

|1〉 →(

01

)

− occupied state, (3.77)


and, analogously for the “bra”-states,

〈0| →(

1 0)

− empty state, (3.78)

〈1| →(

0 1)

− occupied state, (3.79)

where the two form an orthonormal basis with 〈0|0〉 = 〈1|1〉 = 1 and 〈0|1〉 = 0.

Spinor representation of operators

In the spinor representation each second quantization operator becomes a 2×2matrix,

A→(

A00 A01

A10 A11

)

, (3.80)

where Aαβ = 〈α|A|β〉 and α, β = 0, 1.The particle number operator has the following action

n

(

10

)

= 0, (3.81)

n

(

01

)

= 1

(

01

)

, (3.82)

and is, therefore, given by a diagonal matrix in this spinor representation withits eigenvalues on the diagonal,22

n→ 〈n|n|n′〉 = n′1 =

(

0 00 1

)

, (3.83)

and one readily confirms that this is consistent with the action of the operatorgiven by Eqs. (3.81) and (3.82).

Spinor representation of a and a†

Using the definitions (3.60) and (3.61) we readily obtain the matrix elementsof the creation and annihilation operator. We again consider the matrices withrespect to single-particle states |φk〉 and take into account that, for fermions,nk is either 0 or 1. As a result, we obtain

⟨

nk

∣

∣

∣a†k

∣

∣

∣n′k

⟩

=

(

0 01 0

)

= δnk,1δnk,n′

k+1 ≡ A†

k, (3.84)

〈nk |ak|n′k〉 =

(

0 10 0

)

= δnk,0δnk,n′

k−1 ≡ Ak, (3.85)

22The first [second] row corresponds to the case 〈n| = 〈0| [〈n| = 〈1|], whereas the first[second] column corresponds to |n〉 = |0〉 [|n〉 = |1〉].


where the matrix of ak is the transposed of that of a†k and we introduced theshort-hand notation Ak for the matrix δnk,0δn′

k,1 in the space of single-particle

orbitals |φk〉 23.

Matrix elements of a and a† in Fock space

It is now easy to extend this to matrix elements with respect to anti-symmetricN -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 otherorbitals. In addition, there will be a sign factor depending on the position oforbital k in the N -particle state, cf. definitions (3.60) and (3.61),

⟨

{n}∣

∣

∣a†k

∣

∣

∣{n′}

⟩

= (−1)αk δk{n},{n′}A†k (3.86)

where the original prefactor 1 − n′k has been transformed into an additional

Kronecker delta for nk. The matrix of the annihilation operator is

〈{n} |ak| {n′}〉 = (−1)αk δk{n},{n′} Ak (3.87)

Matrix elements of one-particle operators in Fock space

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

23We summarize the main properties of the matrices Ak and A†k which are a consequence

of the properties of a†k and ak and can be verified by direct matrix multiplication:

1. A2k =

(

A†k

)2

= 0.

2. A†kAk =

(

0 00 1

)

= nk1k, – a diagonal matrix with the eigenvalues of nk on the

diagonal, cf. Eq. (3.83).

3. AkA†k =

(

1 00 0

)

= (1− nk)1k = 1−A†kAk, i.e. A†

k and Ak anti-commute.

4. For different single-particle spaces, k 6= l, [A†k,Al]+ = [A†

k,A†l ]+ = [Ak,Al]+ = 0.


and (3.86) for the creator successively we obtain, for the case k 6= l,

⟨

{n}∣

∣

∣a†l ak

∣

∣

∣{n′}

⟩

=∑

{n}

⟨

{n}∣

∣

∣a†l

∣

∣

∣{n}

⟩

〈{n} |ak| {n′}〉 =

= (−1)α′

k

∑

{n}

(−1)αl δnk,0δk{n},{n′}δnk,n

′

k−1δnl,1δ

l{n},{n}δnl+1,nl

= (−1)α′

kδnl,1δkl{n},{n′}

∑

n,nk

(−1)αl δnk,0δnk,nkδnl,n

′

lδnk,n

′

k−1δnl+1,nl

= (−1)αk′lδkl{n},{n′}A†lAk, αk′l =

∑

m<k

n′m +

∑

m<l

nm, (3.88)

which is a diagonal matrix in all orbitals except k and l whereas, with respectto orbital k, it has the structure of the matrix (3.86) and, for orbital l, the formof matrix (3.87). Note that the occupation numbers entering the exponent αk′l

are restricted by the Kronecker symbols. For the case k = l we recover thematrix of the particle number operator which is completely diagonal24

⟨

{n}∣

∣

∣a†kak

∣

∣

∣{n′}

⟩

= 〈{n} |nk| {n′}〉 = nkδ{n},{n′}. (3.89)

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}∣

∣

∣B1

∣

∣

∣{n′}

⟩

=∞∑

k,l=1

blk

⟨

{n}∣

∣

∣a†l ak

∣

∣

∣{n′}

⟩

(3.90)

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

⟨

{n}∣

∣

∣Bdiag

1

∣

∣

∣{n′}

⟩

= δ{n},{n′}

∞∑

k=1

bknk = δ{n},{n′}

N∑

k=1

bnk. (3.91)

where, in the last equality, we have simplified the summation by including onlythe 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 obtainαk′l →

∑

m<k(n′m + nm), δkl{n},{n′} → δk{n},{n′} and the matrix product in k-orbital space

yields A†kAk → nkδnk,n

′

k[cf. property 2 in footnote 23] which combines to the results (3.89).


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

⟨

{n}∣

∣

∣B1

∣

∣

∣{n′}

⟩

=

{

δ{n},{n′}

N∑

k=1

bnknk+

+N∑

k 6=l=1

(−1)k+l+γkl bnlnkδnknl

{n},{n′} A†nlAnk

,

}

. (3.92)

where γkl = 1, for k < l, and 0, otherwise.

Matrix elements of two-particle operators in Fock space

To compute the matrix elements of two-particle operators, Eq. (3.73), we needthe matrix elements of four-operator products, which we transform, using theanti-commutation relations (3.63) according to

a†i a†j alak = −a†i ala†j ak + δjl a

†i ak. (3.93)

Next, transform the matrix element of the first term on the right,⟨

{n}∣

∣

∣a†i ala

†j ak

∣

∣

∣{n′}

⟩

=∑

{n}

⟨

{n}∣

∣

∣a†i al

∣

∣

∣{n}

⟩⟨

{n}∣

∣

∣a†j ak

∣

∣

∣{n′}

⟩

=

=∑

{n}

(−1)αilδil{n},{n}δni,1δni,0δnl,0δnl,1 × (−1)αjk′δjk{n},{n′}δnj ,1δn′

j ,0δnk,0δn′

k,1,

where αjk′ =∑

p<j np+∑

p<k n′ etc. Performing the summation, with the help

of the Kronecker deltas we obtain the final result26⟨

{n}∣

∣

∣a†i ala

†j ak

∣

∣

∣{n′}

⟩

= (−1)αilj′k′δiljk{n},{n′}A†iAlA†

jAk (3.94)

with αilj′k′ =∑

p<i

np +∑

p<l

np +∑

p<j

n′p +

∑

p<k

n′p.

25The non-diagonal matrix elements are transformed to summation over occupied orbitalsas

∞∑

k 6=l=1

blk

⟨

{n}∣

∣

∣a†l ak

∣

∣

∣{n′}

⟩

=N∑

k 6=l=1

bnlnk

⟨

{n}∣

∣a†nlank

∣

∣ {n′}⟩

,

where it remains to carry out the action oft the two operators. Note that the sign of theresult is different for nl < nk and nl > nk.

26We first rewrite∑

{np}

δil{n},{np}δjk

{np},{n′} =∑

ninlnj nk

δiljk

{n},{n′}δnj ,njδnk,nk

δni,n′

iδnl,n

′

l,

Taking into account the other Kronecker deltas we can combine pairs and perform theremaining four summations,

∑

niδni,n

′

iδni,0 = δn′

i,0 and so on.


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.

i=l: (−1)αj′k′δjk{n},{n′}niA†jAk

j=k: (−1)αilδil{n},{n′}njA†iAl

l=j: (−1)αik′δik{n},{n′}(1− nl)A†iAk

i=l, j=k: δ{n},{n′}ninj

k=i: (−1)αlj′δlj{n},{n′}niAlA†j + (−1)αilδil{n},{n′}δijA

†iAl

3.4.3 Fock Matrix of the binary interaction

Of particular importance is the occupation number matrix representation ofthe interaction potential. This is an example of a two-particle quantity theproperties of which we discussed in section 3.4.2. But for this special case, wecan make further progress27. Starting point is the pair interaction

V =1

2

N∑

α 6=β=1

w(α, β), (3.95)

with the second quantization representation (3.73)

V =1

2

∑

ijkl

wijkla†i a

†j alak, (3.96)

where the matrix elements are defined as

wijkl =

∫

d3xd3y φ∗i (x)φ

∗j(y)w(x,y)φk(x)φl(y), (3.97)

and have the following symmetries

wijkl = = wjilk, (3.98)

wijkl = w∗klij, (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.


Eq. (3.96)28

V =∞∑

1≤i<j

∞∑

1≤k<l

w−ijkla

†j a

†i akal, (3.100)

w−ijkl = wijkl − wijlk, (3.101)

where we introduced an antisymmetrized potential w−. Note the change ofthe order of the creation and annihilation operator pairs in Eq. (3.100).

We now compute the matrix of (3.100) with fully antisymmetric vectors|{n}〉 and |{n′}〉

〈{n}|V |{n′}〉 =∞∑

1≤i<j

∞∑

1≤k<l

w−ijkl〈{n}|a

†j a

†i akal|{n′}〉. (3.102)

Each of the two vectors contains N particles (the interaction does not changethe particle number), i.e. exactly N occupied orbitals which are all different.So the sums over i, j and k, l, in fact, run over two (possibly different) sets of Norbitals with the numbers (m1,m2 . . .mN) and (m′

1,m′2 . . .m

′N), respectively,

29

〈{n}|V |{n′}〉 → 〈{m}|V |{m′}〉 =

=N∑

1≤i<j

N∑

1≤k<l

w−mimjm′

km′

l〈{m}|a†mj

a†miam′

kam′

l|{m′}〉. (3.103)

Using the definitions of the creation and annihilation operators, Eqs. (3.60),

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),

∑

kl

wijklalak =∑

k<l

(wijkl − wijlk)alak =∑

k<l

w−ijklalak.

Extending this to the sum over i, j and using the symmetry properties (3.98), we obtain

∑

ij,k<l

(wijkl − wijlk)a†i a

†j alak =

∑

i<j,k<l

(wijkl − wjikl − wijlk + wjilk)a†i a

†j alak =

= 2∑

i<j,k<l

w−ijkla

†i a

†j alak = 2

∑

i<j,k<l

w−ijkla

†j a

†i akal

29by |{m}〉 = |{m}〉(|{n}〉) we will denote the subset of N occupied orbitals containedin the state |{n}〉. For example, a three-particle state |{n}〉 = |1, 0, 0, 1, 1〉 transforms into|m1m2m3〉 where themi point to the original orbitals with numbersm1 = 1,m2 = 4,m3 = 5.Note that the matrix 〈{n}|V |{n′}〉 is diagonal in all orbitals missing simultaneously in 〈{m}|and |{m′}〉.


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

〈{m}|V |{m′}〉 =N∑

1≤i<j

N∑

1≤k<l

(−1)i+j+k+l w−mimjm′

km′

l〈{m}|mi,mj

|{m′}〉m′

k,m′

l,

(3.104)where the notation |{m′}〉m′

k,m′

lmeans that the single-particle orbitals with

number m′k and m′

l are missing in the state |{m′}〉 which now is a state ofN − 2 particles, and similarly for 〈{m}|mi,mj

. The scalar product of the twoanti-symmetric N−2-particle states in (3.104) is non-zero only if the two statescontain N − 2 identical orbitals. To simplify the analysis, in Eq. (3.104) wehave moved the missing orbitals to positions one and two in the states therebyaccumulating the total sign factor contained in this expression. Thus, theremaining orbitals are not only identical but they also have identical numbers,i.e. m3 = m′

3,m4 = m′4, . . . .

Finally, expression (3.104) will be only non-zero if the missing orbitals fallin one of three cases31:

1. The two states are identical, {n} ≡ {n′} and, consequently {m} ≡ {m′}.Then Eq. (3.104) yields

〈{n}|V |{n′}〉 = δ{n},{n′}

N∑

1≤i<j

w−mimjmimj

. (3.105)

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

〈{n}|V |{n′}〉 = δmpm′

r

{n},{n′}A†mp

Am′

r

N−1∑

1≤i,i 6=p,r

(−1)p+r ·Θ(p, r, i)w−mimpmim′

r

(3.106)Here Θ(p, r, i) = −1, if eithermp < mi orm

′r < mi, otherwise Θ(p, r, i) =

1. This case describes single-particle excitations where |{n′}〉 = |{n}rp〉.30Since i < j and k < l, the signs produced by the first and second operators are inde-

pendent of each other.31Thereby we return to the full vectors (including the empty orbitals) and restore the

delta functions.32To obtain the correct sign we move the orbitals p and r to the last place in the product

in state 〈{n}| and in |{n′}〉, respectively and count the number of transpositions (difference).


3. The two states are identical except for two orbitals with the numbers mp

and mq in 〈{m}| and m′r and m

′s in |{m′}〉, respectively. Without loss of

generality we can use mp < mq and m′r < m′

s. Then Eq. (3.104) yields

〈{n}|V |{n′}〉 = δmpmqm′

rm′

s

{n},{n′} A†mp

Am′

rA†

mqAm′

s(−1)p+q+r+sw−

mpmqm′

rm′

s

(3.107)This case describes two-particle excitations where |{n′}〉 = |{n}rspq〉.

These results are known as Slater-Condon rules and were obtained by thosetwo authors in 1929 and 1930 [Sla29, Con30] and are of prime importance forwave function based many-body methods such as configuration interaction(CI) and Multiconfiguration Hartree-Fock (MCHF) and their time-dependentextensions, e.g. [HHB14]. Similarly this representation is used in configura-tion path integral Monte Carlo simulations of strongly correlated fermions,e.g. [SBF+11] and references therein.

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 thatthese considerations are entirely analogous for fermions and bosons and canbe performed at once for both, the only difference being the details of thecommutation (anticommutation) rules of the respective creation and annihila-tion operators. Here we start with the coordinate representation whereas themomentum 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 givenspace point rather than in given orbital φi(r). To this end we consider thesuperposition in terms of the functions φi(r) where the coefficients are thecreation and annihilation operators,

ψ(x) =∞∑

i=1

φi(x)ai, (3.108)

ψ†(x) =∞∑

i=1

φ∗i (x)a

†i . (3.109)

3.5. FIELD OPERATORS 119

Figure 3.3: Illustration of the relation of the field operators to the secondquantization operators defined on a general basis {φi(x)}. The field operatorψ†(x) creates a particle at space point x (in spin state |σ〉) to which all single-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 r, and the φi(x)form a complete orthonormal set. Obviously, these operators have the desiredproperty to create (annihilate) a particle at space point r in spin state σ. Fromthe (anti-)symmetrization properties of the operators a and a† we immediatelyobtain

[

ψ(x), ψ(x′)]

∓= 0, (3.110)

[

ψ†(x), ψ†(x′)]

∓= 0, (3.111)

[

ψ(x), ψ†(x′)]

∓= δ(x− x′). (3.112)

These relations are straightforwardly proven by direct insertion of the def-initions (3.108) and (3.109). We demonstrate this for the last expression.

[

ψ(x), ψ†(x′)]

∓=

∞∑

i,j=1

φi(x)φ∗j(x

′)[

ai, a†j

]

∓=

=∞∑

i=1

φi(x)φ∗i (x

′) = δ(x− x′) = δ(r− r′)δσ,σ′ ,

where, in the last line, we used the representation of the delta function interms of a complete set of functions.


3.5.2 Representation of operators

We now transform operators into second quantization representation using thefield operators, taking advantage of the identical expressions for bosons andfermions.


The general second-quantization representation was given by [cf. Secs. 3.3,3.4]

B1 =∞∑

i,j=1

〈i|b|j〉a†iaj. (3.113)

We now transform the matrix element to coordinate representation:

〈i|b|j〉 =∫

dxdx′φ∗i (x)〈x|b|x′〉φj(x

′), (3.114)

and obtain for the operator, taking into account the definitions (3.108) and(3.109),

B1 =∞∑

i,j=1

∫

dxdx′ a†iφ∗i (x)〈x|b|x′〉φj(x

′)aj =

=

∫

dxdx′ ψ†(x)〈x|b|x′〉ψ(x′). (3.115)

For the important case that the matrix is diagonal, 〈x|b|x′〉 = b(x)δ(x − x′),the final expression simplifies to

B1 =

∫

dx ψ†(x)b(x)ψ(x) (3.116)

Consider a few important examples. The first is again the density operator.In first quantization the operator of the particle density for N particles followsfrom quantizing the classical result for point particles,

n(x) =N∑

α=1

δ(x− xα), (3.117)

3.5. FIELD OPERATORS 121

and the expectation value in a certain N -particle state Ψ(x1, x2, . . . xN) is33

〈n〉(x) = 〈Ψ|N∑

α=1

δ(x− xα)|Ψ 〉

= N

∫

d2d3 . . . dN |Ψ(1, 2, . . . N)|2 = n(r, σ), (3.118)

which is the single-particle spin density of a (in general correlated) N -particlesystem. The second quantization representation of the density operator followsfrom our above result (3.116) by replacing b→ δ(x− x′), i.e.

n(x) =

∫

dx′ψ†(x′)δ(x− x′)ψ(x′) = ψ†(x)ψ(x), (3.119)

and the operator of the total density is the sum (integral) over all coordinate-spin states

N =

∫

dx n(x) =

∫

dx ψ†(x)ψ(x), (3.120)

naturally extending the previous results for a discrete basis to continuousstates.

The second example is the kinetic energy operator which is also diagonaland has the second-quantized representation

T =

∫

dx ψ†(x)

(

−1

2∇2

)

ψ(x). (3.121)

The third example is the second quantization representation of the single-particle potential v(r) given by

V =

∫

dx ψ†(x)v(r)ψ(x). (3.122)


In similar manner we obtain the field operator representation of a generaltwo-particle operator

B2 =1

2

∞∑

i,j,k,l=1

〈ij|b|kl〉a†ia†jalak. (3.123)

33This is the example of an (anti-)symmetrized pure state which is easily extended tomixed states.


We now transform the matrix element to coordinate representation:

〈ij|b|kl〉 =∫

dx1dx2dx3dx4φ∗i (x1)φ

∗j(x2)〈x1x2|b|x3x4〉φl(x3)φk(x4), (3.124)

and, assuming that the matrix is diagonal,〈x1x2|b|x3x4〉 = b(x1, x2)δ(x1 − x3)δ(x2 − x4), we obtain, after inserting thisresult into (3.123),

B2 =1

2

∞∑

i,j,k,l=1

∫

dx1dx2 a†iφ

∗i (x1)a

†jφ

∗j(x2)b(x1, x2)φl(x1)alφk(x2)ak.

Using again the defintion of the field operators the final result for a diagonaltwo-particle operator in coordinate representation is

B2 =1

2

∫

dx1dx2 ψ†(x1)ψ

†(x2)b(x1, x2)ψ(x2)ψ(x1) (3.125)

Note again the inverse ordering of the destruction operators which makes thisresult universally applicable to fermions and bosons. The most importantexample of this representation is the operator of the two-particle interaction,W , which is obtained by replacing b(x1, x2) → w(x1 − x2).

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 theelectron gas or the jellium model, since the eigenfunctions of the momentumoperator,

〈x|φk,s〉 = φk,s(x) =1

V1/2eik·rδs,σ, x = (r, σ), (3.126)

are eigenfunctions of the translation operator. Here we use periodic boundaryconditions to represent an infinite system by a finite box of length L and volumeV = L3, so the wave numbers have discrete values kx = 2πnx/L, . . . kz =2πnz/L with nx, ny, nz being integer numbers. The eigenfunctions (3.126)form a complete orthonormal set, where the orthonormality condition reads

〈φk,s|φk′,s′〉 =1

V

∫

V

d3r ei(k′−k)r

∑

σ

δs,σδs′,σ = δk,k′δs,s′ , (3.127)

where we took into account that the integral over the finite volume V equalszero for k 6= k′ and V otherwise.

3.6. MOMENTUM REPRESENTATION 123

3.6.1 Creation and annihilation operatorsin momentum space

The creation and annihilation operators on the Fock space of N -particle statesconstructed from the orbitals (3.126) are obtained by inverting the definition ofthe field operators (3.108) written with respect to the momentum-spin states(3.126)

ψ(x) =∑

k′σ′

φk′,σ′(x)ak′,σ′ .

Multiplication by φ∗k,σ(x) and integrating over x yields, with the help of con-

dition (3.127),

ak,σ =

∫

dx φ∗k′σ′ψ(x) =

1

V1/2

∫

V

d3r e−ik·rψ(r, σ), (3.128)

and, similarly for the creation operator,

a†k,σ =1

V1/2

∫

V

d3r eik·rψ†(r, σ). (3.129)

Relations (3.128) and (3.129) are nothing but the Fourier transforms of thefield operators. These operators obey the same (anti-)commutation relations asthe 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 momentumspace.


For a single-particle operator we have, according to our general result, Eq.(3.69), and denoting x = (r, s), x′ = (r′, s′),

B1 =∑

kσ

∑

k′σ′

a†kσ〈kσ|b|k′σ′〉 ak′σ′

=∑

kσ

∑

k′σ′

∫

dx dx′a†kσ〈kσ|x〉〈x|b|x′〉〈x′|k′σ′〉 ak′σ′

=1

V∑

kσ

∑

k′σ′

∫

dx dx′a†kσe−ikr〈x|b|x′〉eik′r′ak′σ′δσ,sδσ′,s′ , (3.130)


where, in the last line, we inserted complete sets of momentum eigenstates(3.126). If the momentum matrix elements of the operator b are known, thefirst line can be used directly. Otherwise, the matrix element is obtained fromthe 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 beperformed. We demonstrate this for the example of the kinetic energy operator.Then 〈x|b|x′〉 → −~

2∇2

2mδ(x− x′), and we obtain, using the property (3.127),

T =1

V∑

kσ

∑

k′

∫

V

d3r a†kσ e−ikr ~

2k′2

2meik

′rak′σ

=∑

kσ

~2k2

2ma†kσakσ. (3.131)

In similar fashion we obtain for the single-particle potential, upon replacing〈x|b|x′〉 → v(r)δ(x− x′),

V =∑

kσ

∑

k′

a†kσ ak′σ1

V

∫

V

d3r e−ikr v(r) eik′r

=∑

kσ

∑

k′

vk−k′a†kσ ak′σ, (3.132)

where we introduced the Fourier transform of the single-particle potential,vq = V−1

∫

d3r v(r)e−iqr. Finally, the operator of the single-particle densitybecomes, in momentum space by Fourier transformation,

nq =∑

σ

nqσ =∑

σ

1

V

∫

V

d3r ψ†σ(r)ψσ(r) e

−iqr

=1

V∑

σkk′

a†k′σakσ1

V

∫

V

d3r ei(k−k′)re−iqr

=1

V∑

σk

a†k−q,σakσ. (3.133)

This shows that the Fourier component of the density operator, nq, describesa 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 thesingle-particle potential (3.132) as

V = V∑

q

vq n−q. (3.134)

3.6. MOMENTUM REPRESENTATION 125


We now turn to two-particle operators. Rewriting the general result (3.73) fora spin-momentum basis, we obtain

B2 =1

2!

∑

k1σ1k2σ2

∑

k′

1σ′

1k′

2σ′

2

a†k1σ1a†k2σ2

〈k1σ1k2σ2|b|k′1σ

′1k

′2σ

′2〉 ak′

2σ′

2ak′

1σ′

1(3.135)

We now apply this result to the interaction potential where the matrix elementin momentum representation was computed before, 〈k1σ1k2σ2|w|k′

1σ′1k

′2σ

′2〉 =

w(k1 − k′1)δk1+k2−k′

1−k′

2δσ1,σ′

1δσ2,σ′

2, and w denotes the Fourier transform of the

pair interaction, and the interaction does not change the spin of the involvedparticles, see problem 6, Sec. 3.8. Inserting this into Eq. (3.135) and intro-ducing the momentum transfer q = k′

1 − k1 = k2 − k′2, we obtain

W =1

2!

∑

k1k2q

∑

σ1σ2

w(q)a†k1σ1a†k2σ2

ak2−q,σ2ak1+q,σ1

, (3.136)

In similar manner other two-particle quantities are computed. With thisresult we can write down the second quantization representation in spin-momentum space of a generic hamiltonian that contains kinetic energy, anexternal potential and a pair interaction contribution. From the expressions(3.131, 3.132, 3.136) we obtain

H =∑

kσ

~2k2

2ma†kσakσ +

∑

kk′σ

vk−k′ a†kσak′σ

+1

2!

∑

k1k2q

∑

σ1σ2



. (3.137)

This result is a central starting point for many investigations in condensedmatter 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 spatiallyhomogeneous, so the momentum is conserved. The hamiltonian of this systemfollows from the result (3.137) by omitting the external potential,

H =∑

kσ

~2k2

2ma†kσakσ +N

ξM2

+1

2!

∑

k1k2q

∑

σ1σ2



. (3.138)


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 planewave solutions, we may use the same single-particle orbitals as in the non-relativistic case. With this, the matrix elements of the single-particle potentialand of the interaction potential remain unchanged (if magnetic corrections tothe interaction are neglected). The only change is in the kinetic energy con-tribution, due to the relativistic modification of the single-particle dispersion,ǫk =

~2k2

2m→

√~2k2c2 +m2c4, wherem is the rest mass. In the ultra-relativistic

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 negativeenergy branch ǫk− = −

√~2k2c2 +m2c4 and the corresponding second set of

plane wave states. In all matrix elements we would need to include a secondindex (+,−) referring to the energy band.

3.7 Discussion and outlook

After investigating the basic properties of the method of second quantizationwe now turn to more advanced topics. One of them is the extension of theanalysis to systems at a finite temperature, i.e. in a mixed ensemble. Thiswill be the subject of Chapter 4. After this we turn to the time evolution ofthe field operators following an external perturbation. This will be studied indetail for the case of single-time observables, in Chapter 5. A second route tononequilibrium dynamics is to use field operator products that depend on twotimes which leads to the theory of nonequilibrium Green functions which wediscuss in Chapter 7.

3.8 Problems to Chapter 3

1. Express Λ±123 via Λ±

12, cf. Eqs. (3.11) and (3.12).

2. Generalize the previous result to find a decomposition of Λ±1...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 creationand 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.

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Chapter 3 Fermions and bosons - Uni Kiel