14 1.Second Quantization 1.3.3 General Single-and Many-Particle Operators Let us consider an operator for the N-particle system which is a sum of single-particle operators T=th+t2+.…+tw=厂ta (1.3.16) a are t=(tl》 (1.3.17) such that ttag li)(jl (1.3.18) 汤 and for the full N-particle system T=∑∑l。l。 (1.3.19) a=1 sent this operator in terms of creation and annihilation op tWee by takingpair of states from (1 and cnlcuting their effect on an arbitrary state (1.3.1).We assume initially that ji )aa,n,n3） a =∑)aaS+li,i2,i》Vmm= 1 (1.3.20) =s+∑l间aGl。li,i2iw0mm= 1 It is possible,as was done in the third line,to bring the S+to the front,since it commutes with every symmetric operator.If the state j is nj-fold occupied, it gives rise to nj terms in which Lj)is replaced by li).Hence,the effect of S+is to yield nj states ...,n+1,...,nj-1,...),where the change in the normalization should be noted.Equation (1.3.20)thus leads on to =%vT房%+1…，%-1 =何V4+1,n4+1,…,乃-1，.） (1.3.20 =aal,n,…,nj）
14 1. Second Quantization 1.3.3 General Single- and Many-Particle Operators Let us consider an operator for the N-particle system which is a sum of single-particle operators T = t1 + t2 + ... + tN ≡ α tα , (1.3.16) e.g., for the kinetic energy tα = p2 α/2m, and for the potential V (xα). For one particle, the single-particle operator is t. Its matrix elements in the basis |i are tij = i|t|j, (1.3.17) such that t = i,j tij |i j| (1.3.18) and for the full N-particle system T = i,j tij N α=1 |iα j| α . (1.3.19) Our aim is to represent this operator in terms of creation and annihilation operators. We begin by taking a pair of states i, j from (1.3.19) and calculating their effect on an arbitrary state (1.3.1). We assume initially that j = i α |iα j| α |... ,ni,... ,nj ,... ≡ α |iα j| α S+ |i1, i2,... ,iN 1 √n1!n2! ... = S+ α |iα j| α |i1, i2,... ,iN 1 √n1!n2! ... . (1.3.20) It is possible, as was done in the third line, to bring the S+ to the front, since it commutes with every symmetric operator. If the state j is nj -fold occupied, it gives rise to nj terms in which |j is replaced by |i. Hence, the effect of S+ is to yield nj states |... ,ni + 1,... ,nj − 1,..., where the change in the normalization should be noted. Equation (1.3.20) thus leads on to = nj √ni + 1 1 √nj |... ,ni + 1,... ,nj − 1,... = √nj √ni + 1 |... ,ni + 1,... ,nj − 1,... = a† iaj |... ,ni,... ,nj ,.... (1.3.20 )
1.3B080ns15 Forj=i,the i is replaced ntimes by itself,thus yielding nil....ni..=afail....ni... Thus,for any N,we have 空风侃= From this it follows that,for any single-particle operator, T=∑aa, (1.3.21) where t场=（创t》 (1.3.22) The special caseti=leads to Ho=>eiajai i.e,to(1.3.15a In a similar way one can show that two-particle operators (1.3.23) can be written in the form F-专219kme (1.3.24) where (i.jlf)k.m)=drdyi(z)vj(u)f(,u)p(z)pm(v).(13.25) In (1.3.23),the condition a is required as,otherwise,we would hav only a single-particle operator.The factorin(1.3.23)is to ensure that each interaction is included only once since,for identical particles,symmetry im- plies that f()(xa,xg)=f()(xB,xa). Proof of (1.3.24).One first expresses F in the form F=泛正9k网
1.3 Bosons 15 For j = i, the i is replaced ni times by itself, thus yielding ni |... ,ni,... = a† iai |... ,ni,.... Thus, for any N, we have N α=1 |iα j| α = a† i aj . From this it follows that, for any single-particle operator, T = i,j tija† iaj , (1.3.21) where tij = i|t|j. (1.3.22) The special case tij = iδij leads to H0 = i ia† iai , i.e., to (1.3.15a). In a similar way one can show that two-particle operators F = 1 2 α=β f(2)(xα, xβ) (1.3.23) can be written in the form F = 1 2 i,j,k,m i, j| f(2) |k,m a† i a† jamak, (1.3.24) where i, j| f(2) |k,m = dx dyϕ∗ i (x)ϕ∗ j (y)f(2)(x, y)ϕk(x)ϕm(y) . (1.3.25) In (1.3.23), the condition α = β is required as, otherwise, we would have only a single-particle operator. The factor 1 2 in (1.3.23) is to ensure that each interaction is included only once since, for identical particles, symmetry implies that f(2)(xα, xβ) = f(2)(xβ, xα). Proof of (1.3.24). One first expresses F in the form F = 1 2 α=β i,j,k,m i, j| f(2) |k,m |iα |jβ k| α m| β .
16 1.Second Quantization We now investigate the action of one term of the sum constituting F: l)a》a(a(mlgl,n4,…,n,…,nk…,nm） a≠B 1 =kmm√后打Vn+打 ,n+1,,n+1,,nk-1,.,nm-1,) =afajakam…,n,…,nj…,nk,…,nm） the derivation has to be supplemented to that for the single particle operators. A somewhat shorter derivation,and one which also covers the case of fermions,proceeds as follows:The commutator and anticommutator for bosons and fermions,respectively,are combined in the form [ak,ajl=kj. 三.以风.m-三。.，ma =∑l)。。b)amlg-∑月。ml alaxajam-al lak,ajl am aka干aak =士najakam=aafamak (1.3.26 This completes the proof of the form(1.3.24). 1.4 Fermions 1.4.1 States,Fock Space,Creation and Annihilation Operators For fermions,one needs to consider the states....,iN)defined in (1.2.2),which can also be represented in the form of a determinant: 1 |hi2…i)x s-,2…,iw= (1.4.1) liwh1liw)2…liw）x
16 1. Second Quantization We now investigate the action of one term of the sum constituting F: α=β |iα |jβ k| α m| β |... ,ni,... ,nj ,... ,nk,... ,nm,... = nknm 1 √nk √nm √ni + 1 nj + 1 |... ,ni + 1,... ,nj + 1,... ,nk − 1,... ,nm − 1,... = a† ia† jakam |... ,ni,... ,nj ,... ,nk,... ,nm,... . Here, we have assumed that the states are different. If the states are identical, the derivation has to be supplemented in a similar way to that for the singleparticle operators. A somewhat shorter derivation, and one which also covers the case of fermions, proceeds as follows: The commutator and anticommutator for bosons and fermions, respectively, are combined in the form [ak, aj ]∓ = δkj . α=β |iα |jβ k| α m| β = α=β |iα k| α |jβ m| β = α,β |iα k| α |jβ m| β − k|j δkj α |iα m| α = a† iaka† jam − a† i [ak, a† j ] ∓ aka† j∓a† jak am = ±a† ia† jakam = a† ia† jamak , (1.3.26) for bosons fermions. This completes the proof of the form (1.3.24). 1.4 Fermions 1.4.1 States, Fock Space, Creation and Annihilation Operators For fermions, one needs to consider the states S− |i1, i2,... ,iN defined in (1.2.2), which can also be represented in the form of a determinant: S− |i1, i2,... ,iN = 1 √ N! |i11 |i12 ··· |i1N . . . . . . ... . . . |iN 1 |iN 2 ··· |iN N . (1.4.1)
1.4 Fermions 17 The determinants of one-particle states are called slater determinants.if any of the single-particle states in (1.4.1)are the same,the result is ze ero.This two ide occup antisymmetrized state is normalized to 1.In addition,we have S-li2,i,）=-5-i1,i2,. (1.4.2) This dependence on the order is a general property of determinants. Here,too,we shall characterize the states by specifying their occupation numbers,which can now take the values 0 and 1.The state with n particles in state 1 and n particles in state 2,etc.,is n1,几2，...）. The state in which there are no particles is the vacuum state,represented by 10)=10,0,.). This state must not be confused with the null vector! We combine these states(vacuum state,single-particle states,two-particle states,...to give a state space.In other words,we form the direct sum of the state spaces for the various fixed particle numbers.For fermions,this space is once again known as Fock space.In this state space a scalar product is defined as follows: (m1,2…m1',n2',）=dn1,mdn2,nz…; (1.4.3a) particle number(from a single sub ferent subspaces it always vanishes.Furthermore,we have the completeness relation …ln1,n2,）(1,n2,.=1 (1.4.3b) n1=0n2=0 Here,we wish to introduce creation operators a once again.These must be defined such that the result of applying them twice is zero.Furthermore, the order in which they are applied must play a role.We thus define the creation operators a!by S_lin,i2,...,in)=af af...al 0) (1.4.4) S-l2,1…,w〉=aa…alo) Since these states are equal except in sign,the anticommutator is {a,a}=0, (1.4.5a
1.4 Fermions 17 The determinants of one-particle states are called Slater determinants. If any of the single-particle states in (1.4.1) are the same, the result is zero. This is a statement of the Pauli principle: two identical fermions must not occupy the same state. On the other hand, when all the iα are different, then this antisymmetrized state is normalized to 1. In addition, we have S− |i2, i1,... = −S− |i1, i2,.... (1.4.2) This dependence on the order is a general property of determinants. Here, too, we shall characterize the states by specifying their occupation numbers, which can now take the values 0 and 1. The state with n1 particles in state 1 and n2 particles in state 2, etc., is |n1, n2,.... The state in which there are no particles is the vacuum state, represented by |0 = |0, 0,... . This state must not be confused with the null vector! We combine these states (vacuum state, single-particle states, two-particle states, ... ) to give a state space. In other words, we form the direct sum of the state spaces for the various fixed particle numbers. For fermions, this space is once again known as Fock space. In this state space a scalar product is defined as follows: n1, n2,... |n1 , n2 ,... = δn1,n1δn2,n2 ... ; (1.4.3a) i.e., for states with equal particle number (from a single subspace), it is identical to the previous scalar product, and for states from different subspaces it always vanishes. Furthermore, we have the completeness relation 1 n1=0 1 n2=0 ... |n1, n2,... n1, n2,...| = 11 . (1.4.3b) Here, we wish to introduce creation operators a† i once again. These must be defined such that the result of applying them twice is zero. Furthermore, the order in which they are applied must play a role. We thus define the creation operators a† i by S− |i1, i2,... ,iN = a† i1 a† i2 ...a† iN |0 S− |i2, i1,... ,iN = a† i2 a† i1 ...a† iN |0. (1.4.4) Since these states are equal except in sign, the anticommutator is {a† i , a† j} = 0, (1.4.5a)
18 1.Second Quantization which also implies the impossibility of double occupation ()=0. (1.4.5b tered in (1.4.5a)and the commutator of two {A,B)=A.B+=AB+BA [A.B A.B]=AB-BA (1.4.6) Given these preliminaries,we can now address the precise formulation.If one wants to characterize the states by means of occupation numbers,one has to choose a particular ordering of the states.This is arbitrary but,once chosen, must be adhered to.The states are then represented as m,2=()()”.…0,n4=0,1. (1.4.7 The effect of the operator a must be al.,n4,.）=(1-n(-1)2<.,m+1,.. (1.4.8) The number of particles is increased by 1,but for a state that is already cpied,the factor (1)yields zero.The phase factorc mmutations necessary to bring the a to the position i. The adjoint relation reads: (,n4,a=(1-n)(-1)则(n%+1,. (1.4.9) This yields the matrix element (1.4.10) We now calculate ail,n)=∑Ini)(ndas ln) =ln)(1-n)(-1)zca”y6n+1,m (1.4.11) =(2-n0(-1)<l.,n'-1,n' Here,we have introduced the factor n,since,for ni'=0,the Kronecker delta n=0 always gives zero.The factor ni'also ensures that the right-hand side cannot become equal to the state ...n-1,..)=...,-1,...). To summarize,the effects of the creation and annihilation operators are al.,n4,.）=(1-n)(-1)∑<l.n+1,.) (1.4.12) al.,n,）=n(-1)2<l.,n-1,）
18 1. Second Quantization which also implies the impossibility of double occupation a† i 2 = 0. (1.4.5b) The anticommutator encountered in (1.4.5a) and the commutator of two operators A and B are defined by {A, B} ≡ [A, B]+ ≡ AB + BA [A, B] ≡ [A, B]− ≡ AB − BA . (1.4.6) Given these preliminaries, we can now address the precise formulation. If one wants to characterize the states by means of occupation numbers, one has to choose a particular ordering of the states. This is arbitrary but, once chosen, must be adhered to. The states are then represented as |n1, n2,... = a† 1 n1 a† 2 n2 ... |0 , ni = 0, 1. (1.4.7) The effect of the operator a† i must be a† i |... ,ni,... = (1 − ni)(−1) P j<i nj |... ,ni + 1,.... (1.4.8) The number of particles is increased by 1, but for a state that is already occupied, the factor (1 − ni) yields zero. The phase factor corresponds to the number of anticommutations necessary to bring the a† i to the position i. The adjoint relation reads: ... ,ni,...| ai = (1 − ni)(−1) P j<i nj ... ,ni + 1,...| . (1.4.9) This yields the matrix element ... ,ni,...| ai |... ,ni ,... = (1 − ni)(−1) P j<i nj δni+1,ni . (1.4.10) We now calculate ai |... ,ni ,... = ni |ni ni| ai |ni = ni |ni(1 − ni)(−1) P j<i nj δni+1,ni (1.4.11) = (2 − ni )(−1) P j<i nj |... ,ni − 1,... ni . Here, we have introduced the factor ni , since, for ni = 0, the Kronecker delta δni+1,ni = 0 always gives zero. The factor ni also ensures that the right-hand side cannot become equal to the state |... ,ni − 1,... = |... , −1,.... To summarize, the effects of the creation and annihilation operators are a† i |... ,ni,... = (1 − ni)(−1) P j<i nj |... ,ni + 1,... ai |... ,ni,... = ni(−1) P j<i nj |... ,ni − 1,.... (1.4.12)