1.4 Fermions 19 It follows from this that aa,4,）=(1-n)(-1)2<y(n+1)l.,n4,) =(1-n)l.,ni） (1.4.13a aal.,n4,.）=n(-1)2∑<(1-n4+1)l.,n,.) =n…,ni,, (1.4.13b) since for ni∈{0,1}we have n=niand(-l)2∑<：ny=1.On account one obtains the anticommutator [a,al+=1 In the anticommutator with ij,the phase factor of the two terms is different: [a,a1+x(1-n)n(1-1)=0. Likewise,[i,+for ij,also has different phase factors in the two sum mands a and,sin 一eor文no-）=0 one obtains the following ce aid [a,al+=0,回，al+=0,【a,al+=y (1.4.140 1.4.2 Single-and Many-Particle Operators dllations too.the opbetots cexsctly the same as for poson )erat The important relation ∑)。la=aa (1.4.15) article(and icle without los of ge neralit the arrangement to be i<i<iN.Application of the left-hand side of(1.4.15)gives ∑l间。lnS-,i2,,iw）=S-l间alai,2,,iw） nj(1-n)s-li1.i2.....in) The symbol implies that the state j)is
1.4 Fermions 19 It follows from this that aia† i |... ,ni,... = (1 − ni)(−1)2P j<i nj (ni + 1)|... ,ni,... = (1 − ni)|... ,ni,... (1.4.13a) a† iai |... ,ni,... = ni(−1)2P j<i nj (1 − ni + 1)|... ,ni,... = ni |... ,ni,... , (1.4.13b) since for ni ∈ {0, 1} we have n2 i = ni and (−1)2P j<i nj = 1. On account of the property (1.4.13b) one can regard a† iai as the occupation-number operator for the state |i. By taking the sum of (1.4.13a,b), one obtains the anticommutator [ai, a† i ]+ = 1. In the anticommutator [ai, a† j]+ with i = j, the phase factor of the two terms is different: [ai, a† j]+ ∝ (1 − nj )ni(1 − 1) = 0 . Likewise, [ai, aj ]+ for i = j, also has different phase factors in the two summands and, since aiai |... ,ni,... ∝ ni(ni−1) = 0, one obtains the following anticommutation rules for fermions: [ai, aj]+ = 0, [a† i , a† j ]+ = 0, [ai, a† j ]+ = δij . (1.4.14) 1.4.2 Single- and Many-Particle Operators For fermions, too, the operators can be expressed in terms of creation and annihilation operators. The form is exactly the same as for bosons, (1.3.21) and (1.3.24). Now, however, one has to pay special attention to the order of the creation and annihilation operators. The important relation X α |iα j| α = a† iaj , (1.4.15) from which, according to (1.3.26), one also obtains two-particle (and many-particle) operators, can be proved as follows: Given the state S− |i1, i2,... ,iN , we assume, without loss of generality, the arrangement to be i1 < i2 < ... < iN . Application of the left-hand side of (1.4.15) gives X α |iα j| α S− |i1, i2,... ,iN = S− X α |iα j| α |i1, i2,... ,iN = nj (1 − ni)S− |i1, i2,... ,iN ˛ ˛ j→i . The symbol |j→i implies that the state |j is replaced by |i. In order to bring the i into the right position, one has to carry out P k<j nk + P k<i nk permutations of rows for i ≤ j and P k<j nk + P k<i nk − 1 permutations for i>j
20 1.Second Quantization This yields the same phase factor as does the right-hand side of(1.4.15): aal,n,,n）=n(-1)∑.…,n,n-1, =n(1-4)(-1)∑k<nk+∑k<k-64>1l.,n+1,.,n-1, In summary,for bosons and fermions,the single-and two-particle operators can be written,respectively,as T=∑ta (1.4.16a F=2∑(，jf1k,m）aajamak, (1.4.16b) i.j.k,m where the operators a obey the commutation relations(1.3.7)for bosons and,for fermions,the anticommutation relations (1.4.14).The Hamiltonian of a many-particle system with kinetic energy T,potential energy U and a two-particle interaction f(2)has the form =(tij+Uij)ajaj+5 2 (1,m)aiajamak (1.4.16c where the matrix elements are defined in (1.3.21,1.3.22,1.3.25)and,for fermions,particular attention must be paid to the order of the two annihila- tion operators in the two-particle operator. From this point on,the development of the theory can be presented simulta- neously for boson and fermior 1.5 Field Operators 1.5.1 Transformations Between Different Basis Systems Consider two basis systems{i)}and {))What is the relationship between the operators a;and ax? The state A)can be expanded in the basis {li)): )=∑)() (1.5.1) The operator a creates particles in the state).Hence,the superposition a yields one particle in the state )This leads to the relation a=∑Wa (1.5.2a
20 1. Second Quantization This yields the same phase factor as does the right-hand side of (1.4.15): a† iaj |... ,ni,... ,nj ,... = nj (−1) P k<j nk a† i |... ,ni,... ,nj − 1,... = ni(1 − ni)(−1) P k<i nk+P k<j nk−δi>j |... ,ni + 1,... ,nj − 1,... . In summary, for bosons and fermions, the single- and two-particle operators can be written, respectively, as T = i,j tija† i aj (1.4.16a) F = 1 2 i,j,k,m i, j| f(2) |k,m a† ia† jamak, (1.4.16b) where the operators ai obey the commutation relations (1.3.7) for bosons and, for fermions, the anticommutation relations (1.4.14). The Hamiltonian of a many-particle system with kinetic energy T , potential energy U and a two-particle interaction f(2) has the form H = i,j (tij + Uij )a† iaj + 1 2 i,j,k,m i, j| f(2) |k,m a† i a† jamak , (1.4.16c) where the matrix elements are defined in (1.3.21, 1.3.22, 1.3.25) and, for fermions, particular attention must be paid to the order of the two annihilation operators in the two-particle operator. From this point on, the development of the theory can be presented simultaneously for bosons and fermions. 1.5 Field Operators 1.5.1 Transformations Between Different Basis Systems Consider two basis systems {|i} and {|λ}. What is the relationship between the operators ai and aλ? The state |λ can be expanded in the basis {|i}: |λ = i |i i|λ. (1.5.1) The operator a† i creates particles in the state |i. Hence, the superposition ii|λ a† i yields one particle in the state |λ. This leads to the relation a† λ = i i|λ a† i (1.5.2a)
1.5 Field Operators 21 with the adjoint a=∑a (1.5.2b) The position eigenstates x)represent an important special case x)=9(x): (1.5.3) Where x is the single-particle wave function in the coordinate representa creation and a oflddtotete 1.5.2 Field Operators The field operators are defined by tx)=∑p(xa (1.5.4a) tx)=∑px)a (1.5.4b) annihilates)a particle in the commutation relations: b(x),(x)]t=0, (1.5.5a [wt(x),t(x】±=0 (1.5.5b) w(x),6tx±=∑p(x)exa,l± (1.5.5c P(x)p(x)6=63(x-x) where the upper sign applies to fermions and the lower one to bosons We shall now express a few important operators in terms of the field operators. Kinetic energy' 2 2m drV(x)Vu(x) (1.5.6a) 7 The second line in (1.5.6a)holds when the which the ator acts decreases sufficiently fast at infinity that one can neglect the surface contribution to the partial integration
1.5 Field Operators 21 with the adjoint aλ = i λ|i ai. (1.5.2b) The position eigenstates |x represent an important special case x|i = ϕi(x), (1.5.3) where ϕi(x) is the single-particle wave function in the coordinate representation. The creation and annihilation operators corresponding to the eigenstates of position are called field operators. 1.5.2 Field Operators The field operators are defined by ψ(x) = i ϕi(x)ai (1.5.4a) ψ†(x) = i ϕ∗ i (x)a† i . (1.5.4b) The operator ψ†(x) (ψ(x)) creates (annihilates) a particle in the position eigenstate |x, i.e., at the position x. The field operators obey the following commutation relations: [ψ(x), ψ(x )]± = 0 , (1.5.5a) [ψ†(x), ψ†(x )]± = 0 , (1.5.5b) [ψ(x), ψ†(x )]± = i,j ϕi(x)ϕ∗ j (x )[ai, a† j ]± (1.5.5c) = i,j ϕi(x)ϕ∗ j (x )δij = δ(3)(x − x ) , where the upper sign applies to fermions and the lower one to bosons. We shall now express a few important operators in terms of the field operators. Kinetic energy7 i,j a† iTijaj = i,j d3x a† iϕ∗ i (x) − 2 2m ∇2 ϕj (x)aj = 2 2m d3x ∇ψ†(x)∇ψ(x) (1.5.6a) 7 The second line in (1.5.6a) holds when the wave function on which the operator acts decreases sufficiently fast at infinity that one can neglect the surface contribution to the partial integration.
2 1.Second Quantization Single-particle potential ∑i0,4=∑raei(x0x)9,xa =PU(x)vi(x)v(x) (1.5.6b) Two-particle interaction or any two-particle operator 专∑.∫P-P/pi()-j(x)V(x.x)ox(x)pm()ojajumu ik.m -/PP/v.ai Hamiltonian H=/r:(会V+UWee地）+ /rrpipitxvkxexey (1.5.6d) Particle density (particle-number density) The particle-density operator is given by n(x)=63(x-xa). (1.5.7) a Hence its representation in terms of creation and annihilation operators is n(x)=aa ui(y)6)(x-y)v;(y) =∑alajvi(x)px (1.5.8) This representation is valid in any basis and can also be expressed in terms of the field operators n(x)=v(x)v(x) (1.5.9) Total-particle-number operator N=drn(x)=r(x)(x) (1.5.10) Formally,at least,the particle-density operator (1.5.9)of the many- particle system looks like the probability density of a particle in the state (x).However,the analogy is no more than a formal one since the former is and the latte omplex funct This form tion operator formi c b a c nal c to the ,since
22 1. Second Quantization Single-particle potential i,j a† iUijaj = i,j d3x a† iϕ∗ i (x)U(x)ϕj (x)aj = d3x U(x)ψ†(x)ψ(x) (1.5.6b) Two-particle interaction or any two-particle operator 1 2 i,j,k,m d3xd3x ϕ∗ i (x)ϕ∗ j (x )V (x, x )ϕk(x)ϕm(x )a† ia† jamak = 1 2 d3xd3x V (x, x )ψ† (x)ψ† (x )ψ(x )ψ(x) (1.5.6c) Hamiltonian H = d3x 2 2m ∇ψ†(x)∇ψ(x) + U(x)ψ†(x)ψ(x) + 1 2 d3xd3x ψ†(x)ψ†(x )V (x, x )ψ(x )ψ(x) (1.5.6d) Particle density (particle-number density) The particle-density operator is given by n(x) = α δ(3)(x − xα) . (1.5.7) Hence its representation in terms of creation and annihilation operators is n(x) = i,j a† iaj d3y ϕ∗ i (y)δ(3)(x − y)ϕj (y) = i,j a† iajϕ∗ i (x)ϕj (x). (1.5.8) This representation is valid in any basis and can also be expressed in terms of the field operators n(x) = ψ† (x)ψ(x). (1.5.9) Total-particle-number operator Nˆ = d3x n(x) = d3x ψ†(x)ψ(x) . (1.5.10) Formally, at least, the particle-density operator (1.5.9) of the manyparticle system looks like the probability density of a particle in the state ψ(x). However, the analogy is no more than a formal one since the former is an operator and the latter a complex function. This formal correspondence has given rise to the term second quantization, since the operators, in the creation and annihilation operator formalism, can be obtained by replacing the
1.5 Field Operators 23 wave function)in the single-particle densities by the operator).This immediately enables one to write down,e.g.,the curren -density operato (see Problem 1.6) j(x)=2m)( (1.5.11) of the kinetic f the number density dc3g()(6(x-El)()=(x)p(x) (1.5.12) In general,for a k-particle operatorVi: ee d...Edt'...'v()...(E) (5152.5ki3.)().p(i) (1.5.13) 1.5.3 Field Equations The equations of motion of the field operators(x,t)in the Heisenberg rep- resentation (x,)=e4h(x,0)e-ih (1.5.14) read,for the Hamiltonian (1.5.6d). h2 品6x=(（益+U）x)+ +V(x.x(x). (1.5.15) The structure is that of a nonlinear Schrodinger equation,another reason for e te o o mt ih元x)=-H,(x,t=-ehH,(x0e-ih. (1.5.16) Using the relation aBq-=AB,G4E年AC:BE (1.5.17)
1.5 Field Operators 23 wave function ψ(x) in the single-particle densities by the operator ψ(x). This immediately enables one to write down, e.g., the current-density operator (see Problem 1.6) j(x) = 2im[ψ†(x)∇ψ(x) − (∇ψ† (x))ψ(x)] . (1.5.11) The kinetic energy (1.5.12) has a formal similarity to the expectation value of the kinetic energy of a single particle, where, however, the wavefunction is replaced by the field operator. Remark. The representations of the operators in terms of field operators that we found above could also have been obtained directly. For example, for the particlenumber density Z d3 ξd3 ξ ψ† (ξ)ξ| δ(3)(x − bξ) ˛ ˛ξ ¸ ψ(ξ ) = ψ† (x)ψ(x), (1.5.12) where bξ is the position operator of a single particle and where we have made use of the fact that the matrix element within the integral is equal to δ(3)(x−ξ)δ(3)(ξ−ξ ). In general, for a k-particle operator Vk: Z d3 ξ1 ...d3 ξkd3 ξ1 ...d3 ξk ψ† (ξ1)...ψ† (ξk) ξ1ξ2 ... ξk| Vk ˛ ˛ξ 1ξ 2 ... ξ k ¸ ψ(ξ k)...ψ(ξ 1). (1.5.13) 1.5.3 Field Equations The equations of motion of the field operators ψ(x, t) in the Heisenberg representation ψ(x, t)=eiHt/ ψ(x, 0) e−iHt/ (1.5.14) read, for the Hamiltonian (1.5.6d), i ∂ ∂tψ(x, t) = − 2 2m ∇2 + U(x) ψ(x, t) + + d3x ψ†(x , t)V (x, x )ψ(x , t)ψ(x, t). (1.5.15) The structure is that of a nonlinear Schr¨odinger equation, another reason for using the expression “second quantization”. Proof: One starts from the Heisenberg equation of motion i ∂ ∂tψ(x, t) = −[H, ψ(x, t)] = −eiHt/ [H, ψ(x, 0)] e−iHt/ . (1.5.16) Using the relation [AB, C]− = A[B,C]± ∓ [A, C]±B Fermi Bose , (1.5.17)