1.2 Completely Symmetric and Antisymmetric States 9 space of N-particle states.The symmetrized and antisymmetrized basis states are then defined by 54i,2…w=而2分 (±1)PPi,i2,…,iw）· (1.2.2) In other words,we apply all N!elements of the permutation group SN of N objects and,for fermions,we multiply by (-1)when P is an odd permutation. The states defined in (1.2.2)are of two types:completely symmetric and completely antisymmetric. Remarks regarding the properties of S.±≡∑p(仕l)PP (i)Let SN be the permutation group (or symmetric group)of N quantities. Assertion:For every element PE SN,one has PSN =SN. Proof.The set PSN contains exactly the same number of elements as SN and these PSredifterent since if ome hadPtherore,the elemc due to the gr Thus PSN SNP=SN. (1.2.3) (ii)It follows from this that PS+=S+P=S4 (1.2.4a and PS-=S-P=(-1)PS_- (1.2.4b) If P is even,then even elements remain even and odd ones remain odd.If P is odd,then multiplication by P changes even into odd elements and vice versa. PS+lin,...,in)=S+lin,...,iN) PS-li1,.,iw)=(-1)PS_-li1,.,iw） Special case PiS-li1,…,iw)=-S-li1,…,iN） (iii)If,single-particle states occ rring more than once )isno longer normalized to unity.Let us assume that th first state occurs ni times,the second n2 times,etc.Since S+li1,...,iN) contains a total of N!terms,of which rms occurs with a multiplicity of ni!n2!acor the 1 NI 1,…,iwSS+1,,iw= mn2..)2 1n2.. =n1n2
1.2 Completely Symmetric and Antisymmetric States 9 space of N-particle states. The symmetrized and antisymmetrized basis states are then defined by S± |i1, i2,... ,iN ≡ 1 √ N! P (±1)P P |i1, i2,... ,iN . (1.2.2) In other words, we apply all N! elements of the permutation group SN of N objects and, for fermions, we multiply by (−1) when P is an odd permutation. The states defined in (1.2.2) are of two types: completely symmetric and completely antisymmetric. Remarks regarding the properties of S± ≡ √ 1 N! P (±1)P P: (i) Let SN be the permutation group (or symmetric group) of N quantities. Assertion: For every element P ∈ SN , one has P SN = SN . Proof. The set P SN contains exactly the same number of elements as SN and these, due to the group property, are all contained in SN . Furthermore, the elements of P SN are all different since, if one had P P1 = P P2, then, after multiplication by P −1, it would follow that P1 = P2. Thus P SN = SN P = SN . (1.2.3) (ii) It follows from this that P S+ = S+P = S+ (1.2.4a) and P S− = S−P = (−1)P S−. (1.2.4b) If P is even, then even elements remain even and odd ones remain odd. If P is odd, then multiplication by P changes even into odd elements and vice versa. P S+ |i1,... ,iN = S+ |i1,... ,iN P S− |i1,... ,iN = (−1)P S− |i1,... ,iN Special case PijS− |i1,... ,iN = −S− |i1,... ,iN . (iii) If |i1,... ,iN contains single-particle states occurring more than once, then S+ |i1,... ,iN is no longer normalized to unity. Let us assume that the first state occurs n1 times, the second n2 times, etc. Since S+ |i1,... ,iN contains a total of N! terms, of which N! n1!n2!... are different, each of these terms occurs with a multiplicity of n1!n2! ... . i1,... ,iN | S† +S+ |i1,... ,iN = 1 N! (n1!n2! ...) 2 N! n1!n2! ... = n1!n2! ...
10 1.Second Quantization Thus,the normalized Bose basis functions are 1 si7mm=Nmm于PhiW.12.司 (iv)A further property of S is S￥=VWS+ (1.2.6a) since S星=办∑p(t1)PPS±=太∑pS4=vmS±.We now consider an arbitrary N-particle state,which we expand in the basis li)...liN) )=).iw)1,,iwl2) ClN Application of S yields S4l)=∑S4l)lw)e…w=∑)liw)5t4w i,....IN and further application ofS±，with the identity(（2.6a),results in S±l)= 贡，∑S生wS4w (1.2.6b) 1.3 Bosons 1.3.1 States,Fock Space,Creation and Annihilation Operators The state(1.2.5)is fully characterized by specifying the occupation numbers 1 lm1,n2,）=S4l1,i2…,iw） Vniln2!... (1.3.1) Here,n is the number of times that the state 1 occurs,n2 the number of times that state 2 occurs,....Alternatively:m is the number of particles in state 1,n2 is the number of particles in state 2,....The sum of all occupation numbers n must be equal to the total number of particles: (1.3.2)
10 1. Second Quantization Thus, the normalized Bose basis functions are S+ |i1,... ,iN 1 √n1!n2! ... = 1 √N!n1!n2! ... P P |i1,... ,iN . (1.2.5) (iv) A further property of S± is S2 ± = √ N!S± , (1.2.6a) since S2 ± = √ 1 N! P (±1)P P S± = √ 1 N! P S± = √ N!S±. We now consider an arbitrary N-particle state, which we expand in the basis |i1... |iN |z = i1,... ,iN |i1... |iN i1,... ,iN |z ci1,... ,iN . Application of S± yields S± |z = i1,... ,iN S± |i1... |iN ci1,... ,iN = i1,... ,iN |i1... |iN S±ci1,... ,iN and further application of √ 1 N! S±, with the identity (1.2.6a), results in S± |z = 1 √ N! i1,... ,iN S± |i1... |iN (S±ci1,... ,iN ). (1.2.6b) Equation (1.2.6b) implies that every symmetrized state can be expanded in terms of the symmetrized basis states (1.2.2). 1.3 Bosons 1.3.1 States, Fock Space, Creation and Annihilation Operators The state (1.2.5) is fully characterized by specifying the occupation numbers |n1, n2,... = S+ |i1, i2,... ,iN 1 √n1!n2! ... . (1.3.1) Here, n1 is the number of times that the state 1 occurs, n2 the number of times that state 2 occurs, ... . Alternatively: n1 is the number of particles in state 1, n2 is the number of particles in state 2, ... . The sum of all occupation numbers ni must be equal to the total number of particles: ∞ i=1 ni = N. (1.3.2)
1.3 Bosons 11 Apart from this constraint,the ni can take any of the values 0,1,2,. The factor (n.)-1/2,together r with the factor 1/contained in ect of normalizing n. (ii)).Th et st of complt symetreparticle stat By linea superposition,one can construct from these any desired symmetric N-particle state. We now combine the states for N=0,1,2,...and obtain a complete orthonormal system of states for arbitrary particle number,which satisfy the orthogonality relation (1,n2,.n1',n2',.）=dn1n1dn2,n2… (1.3.3a and the completeness relation ∑lm1,2,.)(m1,n2=1 (1.3.3b n1,n2 This extended space is the direct sum of the space with no particles(vacuum state l0)).the space with one particle.the space with two particles.etc.:it is space we have considered so far act only within a subspa ce of again an -particle state.We now d ew obtain which lead from the space of N-particle states to the spaces of N+1-particle states: al,n,）=m+,n+1,. (1.3.40 Taking the adjoint of this equation and relabeling nn,we have (.,n',.a=Vn'+1(.,n'+1,.. (1.3.5) Multiplying this equation by ...n.)yields (,n',al.,n,）=√dn+1,n4 Expressed in words,the operator ai reduces the occupation number by 1. Assertion: a.,n4,)=V而…，n4-1,）forn≥1 (1.3.6) and al…n=0）=0 6 In the states the scalar product between states of differing particle number is defined by (1.3.3a)
1.3 Bosons 11 Apart from this constraint, the ni can take any of the values 0, 1, 2,... . The factor (n1!n2! ...)−1/2, together with the factor 1/ √ N! contained in S+, has the effect of normalizing |n1, n2,... (see point (iii)). These states form a complete set of completely symmetric N-particle states. By linear superposition, one can construct from these any desired symmetric N-particle state. We now combine the states for N = 0, 1, 2,... and obtain a complete orthonormal system of states for arbitrary particle number, which satisfy the orthogonality relation6 n1, n2,... |n1 , n2 ,... = δn1,n1δn2,n2 ... (1.3.3a) and the completeness relation n1,n2,... |n1, n2,... n1, n2,...| = 11 . (1.3.3b) This extended space is the direct sum of the space with no particles (vacuum state |0), the space with one particle, the space with two particles, etc.; it is known as Fock space. The operators we have considered so far act only within a subspace of fixed particle number. On applying p, x etc. to an N-particle state, we obtain again an N-particle state. We now define creation and annihilation operators, which lead from the space of N-particle states to the spaces of N ±1-particle states: a† i |... ,ni,... = √ni + 1 |... ,ni + 1,.... (1.3.4) Taking the adjoint of this equation and relabeling ni → ni , we have ... ,ni ,...| ai = ni + 1 ... ,ni + 1,...| . (1.3.5) Multiplying this equation by |... ,ni,... yields ... ,ni ,...| ai |... ,ni,... = √ni δni+1,ni . Expressed in words, the operator ai reduces the occupation number by 1. Assertion: ai |... ,ni,... = √ni |... ,ni − 1,... for ni ≥ 1 (1.3.6) and ai |... ,ni = 0,... = 0 . 6 In the states |n1, n2,..., the n1, n2 etc. are arbitrary natural numbers whose sum is not constrained. The (vanishing) scalar product between states of differing particle number is defined by (1.3.3a).
12 1.Second Quantization Proof: 0 元l.…,n4-1,.)forn4≥1 30 for n=0' The operator a!increases the occupation number of the state i)by 1,and the operator ai reduces it by 1.The operators a!and ai are thus called creation and annihilation operators.The above relations and the completeness of the states yield the Bose [a,al=0,[a,a】=0,a,a=d (1.3.7a,b,c Proof.It is clear that (1.3.7a)holds for i=j,since a commutes with itself.For ij,it follows from (1.3.6)that aa.,n）=√元l.,n-1,.,n%-1,》 =ajal.,n4,,n,） d,ing he () aa,n,,n,.=V元Vn+,n-1,n+1,) =aa…n…… and (aa{-aa)l…,n,n〉= (,+IVm+I-mml.,,,,) hence also proving(1.3.7c). Starting from the ground state vacuum state 10=0,0.), (1.3.8) which contains no particles at all,we can construct all states: single-particle states a10),., two-particle states a)m.m…
12 1. Second Quantization Proof: ai |... ,ni,... = ∞ ni=0 |... ,ni ,... ... ,ni ,...| ai |... ,ni,... = ∞ ni=0 |... ,ni ,... √ni δni+1,ni = √ni |... ,ni − 1,... for ni ≥ 1 0 for ni = 0 . The operator a† i increases the occupation number of the state |i by 1, and the operator ai reduces it by 1. The operators a† i and ai are thus called creation and annihilation operators. The above relations and the completeness of the states yield the Bose commutation relations [ai, aj]=0, [a† i , a† j ]=0, [ai, a† j] = δij . (1.3.7a,b,c) Proof. It is clear that (1.3.7a) holds for i = j, since ai commutes with itself. For i = j, it follows from (1.3.6) that aiaj |... ,ni,... ,nj ,... = √ni √nj |... ,ni − 1,... ,nj − 1,... = ajai |... ,ni,... ,nj ,... which proves (1.3.7a) and, by taking the hermitian conjugate, also (1.3.7b). For j = i we have aia† j |... ,ni,... ,nj ,... = √ni pnj + 1 |... ,ni − 1,... ,nj + 1,... = a† jai |... ,ni,... ,nj ,... and “ aia† i − a† iai ” |... ,ni,... ,nj ,... = `√ni + 1√ni + 1 − √ni √ni ´ |... ,ni,... ,nj ,... hence also proving (1.3.7c). Starting from the ground state ≡ vacuum state |0≡|0, 0,..., (1.3.8) which contains no particles at all, we can construct all states: single-particle states a† i |0,... , two-particle states 1 √ 2! a† i 2 |0, a† ia† j |0,...
1.3B080ns13 and the general many-particle state a=ma=(a)(回)”m (1.3.9) Normalization: atln-1)=√元lnl (1.3.10) llat In-1)l =v =左aa-) 1.3.2 The Particle-Number Operator The particle-number operator (occupation-number operator for the state)) is defined by (1.3.11) The states introduced above are eigenfunctions of il…,n,=nh,n,） (1.3.12) and the corresponding eigenvalue of is the number of particles in the state The operator for the total number of particles is given by N=∑ (1.3.13) Applying this operator to the states ....yields N1m1,n2,=（ (1.3.14) Assuming that the particles do not interact with one another and,further- more,that the states i)are the eigenstates of the single-particle Hamiltonian with eigenvalues,the full Hamiltonian can be written as Ho=fuei (1.3.15a) Hon1,.）= Snei li...) (1.3.15b) The commutation relations and the properties of the particle-number opera- tor are analogous to those of the harmonic oscillator
1.3 Bosons 13 and the general many-particle state |n1, n2,... = 1 √n1!n2! ... a† 1 n1 a† 2 n2 ... |0. (1.3.9) Normalization: a† |n − 1 = √n |n (1.3.10) a† |n − 1 = √n |n = 1 √n a† |n − 1 . 1.3.2 The Particle-Number Operator The particle-number operator (occupation-number operator for the state |i) is defined by nˆi = a† i ai . (1.3.11) The states introduced above are eigenfunctions of ˆni: nˆi |... ,ni,... = ni |... ,ni,..., (1.3.12) and the corresponding eigenvalue of ˆni is the number of particles in the state i. The operator for the total number of particles is given by Nˆ = i nˆi. (1.3.13) Applying this operator to the states |... , nˆi,... yields Nˆ |n1, n2,... = i ni |n1, n2,.... (1.3.14) Assuming that the particles do not interact with one another and, furthermore, that the states |i are the eigenstates of the single-particle Hamiltonian with eigenvalues i, the full Hamiltonian can be written as H0 = i nˆii (1.3.15a) H0 |n1,... = i nii |n1,.... (1.3.15b) The commutation relations and the properties of the particle-number operator are analogous to those of the harmonic oscillator