4 1.Second Quantization P(,i,,j)=(…j…,i,…） (1.1.3) We remind the reader of a few important properties of this operator.Since P=1,the eiger of Dne to the setry of the Hamilto PH=HP. (1.1.40 The permutation group SN which consists of all permutations of N objects has N!elements.Every permutation P can be represented as a product of transpositions Pij.An element is said to be even (odd)when the number of P;'s is even (odd).2 A few properties: (i)If (1,...,N)is an eigenfunction of H with eigenvalue E,then the same also holds true for Pu(1,...,N). Proof.H地=E地→HP地=PH地=EP业 (ii)For every permutation one has ()=(PlP）, (1.1.5) as follows by renaming the integration variables. (iii)The adjoint permutation operator Pr is defined as usual by lP)=(ptolu〉 It follows from this that (P)=(P-1P-1P〉=(P-1lp〉→pt=p-1 and thus P is unitary PtP=PPt =1. (1.1.6) (iv)For every symmetric operator S(1,...,N)we have [P,S=0 (1.1.7 and (PSlP〉=(PISP》=(PPSl〉=(Sl (1.1.8) This proves that the matrix elements of symmetric operators are the same in the states and in the permutated states P 2 It is well kno n that e that have noelement como()(35)Every ycle can be write a product of transpositions. A=8=a42e
4 1. Second Quantization Pijψ(. . . , i, . . . , j, . . .) = ψ(. . . , j, . . . , i, . . .). (1.1.3) We remind the reader of a few important properties of this operator. Since P2 ij = 1, the eigenvalues of Pij are ±1. Due to the symmetry of the Hamiltonian, one has for every element P of the permutation group P H = HP. (1.1.4) The permutation group SN which consists of all permutations of N objects has N! elements. Every permutation P can be represented as a product of transpositions Pij . An element is said to be even (odd) when the number of Pij ’s is even (odd).2 A few properties: (i) If ψ(1,... ,N) is an eigenfunction of H with eigenvalue E, then the same also holds true for P ψ(1,... ,N). Proof. Hψ = Eψ ⇒ HPψ = PHψ = EPψ . (ii) For every permutation one has ϕ|ψ = P ϕ|P ψ , (1.1.5) as follows by renaming the integration variables. (iii) The adjoint permutation operator P† is defined as usual by ϕ|P ψ = P†ϕ|ψ . It follows from this that ϕ|P ψ = P −1ϕ|P −1P ψ = P −1ϕ|ψ ⇒ P† = P −1 and thus P is unitary P† P = P P† = 1 . (1.1.6) (iv) For every symmetric operator S(1,... ,N) we have [P, S] = 0 (1.1.7) and P ψi| S |P ψj = ψi| P†SP |ψj = ψi|P†P S |ψj = ψi| S |ψj . (1.1.8) This proves that the matrix elements of symmetric operators are the same in the states ψi and in the permutated states P ψi. 2 It is well known that every permutation can be represented as a product of cycles that have no element in common, e.g., (124)(35). Every cycle can be written as a product of transpositions, e.g. (12) odd P124 ≡ (124) = (14)(12) even Each cycle is carried out from left to right (1 → 2, 2 → 4, 4 → 1), whereas the products of cycles are applied from right to left.
1.1 Identical Particles,Many-Particle States,and Permutation Symmetry 5 (v)The converse of (iv)is also true.The requirement that an exchange of articles should not have any observable cor es implies that all mu e symm oni OPl)holds for arbitrary v. 'Thus.POP O and,hence,PO=OP Since identical particles are all influenced identically by any physical pro- cess,all physical operators must be symmetric.Hence,the states and P are experimentally indistinguishable.The question arises as to whether all these N!states are realized in nature In fact the totally s symmetric and totally antisymmetric states and do play a special role.These state e defi ned by P:(,i,…i,…）=:(…,i,…,i,…） (1.1.9) for all Pi It.is an eperimental fact that there are two types of particle,boson and fermion whose stat s are totally sy etri and totally ant symmetric mentioned at the outset,bosons have integral,and fermions Remarks: (i)The symmetry character of a state does not change in the course of time: 0)→P= P(0) (1.1.10) where T is the time-ordering operator.3 (ii)For arbitrary permutations P,the states introduced in(1.1.9)satisfy P中。=。 (1.1.11) Pva=(-1)Pva,with (-1)P= 1 for even permutations -1for odd permutations Thus,the states and a form the basis of two one-dimensional repre sentations of the permutation group SN.For v.,every P is assigned the number 1,and for va every even (odd)element is assigned the number 1(-1).Since,in the case of three or more particles,the P;do not all com- mute with one another,there are,in addition to and a,also states not al P are al.Due to functions of all P cannc Th ese stat are basis fun ns of higher-dimens ns of th tation group.These states are not realized in nature;they are referred to 3 QM I Chap.16
1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 5 (v) The converse of (iv) is also true. The requirement that an exchange of identical particles should not have any observable consequences implies that all observables O must be symmetric, i.e., permutation invariant. Proof. ψ| O |ψ = P ψ| O |P ψ = ψ| P†OP |ψ holds for arbitrary ψ. Thus, P†OP = O and, hence, P O = OP. Since identical particles are all influenced identically by any physical process, all physical operators must be symmetric. Hence, the states ψ and P ψ are experimentally indistinguishable. The question arises as to whether all these N! states are realized in nature. In fact, the totally symmetric and totally antisymmetric states ψs and ψa do play a special role. These states are defined by Pijψ s a (. . . , i, . . . , j, . . .) = ±ψ s a (. . . , i, . . . , j, . . .) (1.1.9) for all Pij . It is an experimental fact that there are two types of particle, bosons and fermions, whose states are totally symmetric and totally antisymmetric, respectively. As mentioned at the outset, bosons have integral, and fermions half-integral spin. Remarks: (i) The symmetry character of a state does not change in the course of time: ψ(t) = T e − i Rt 0 dt H(t ) ψ(0) ⇒ P ψ(t) = T e − i Rt 0 dt H(t ) P ψ(0) , (1.1.10) where T is the time-ordering operator.3 (ii) For arbitrary permutations P, the states introduced in (1.1.9) satisfy P ψs = ψs (1.1.11) P ψa = (−1)P ψa , with (−1)P = 1 for even permutations −1 for odd permutations. Thus, the states ψs and ψa form the basis of two one-dimensional representations of the permutation group SN . For ψs, every P is assigned the number 1, and for ψa every even (odd) element is assigned the number 1(−1). Since, in the case of three or more particles, the Pij do not all commute with one another, there are, in addition to ψs and ψa, also states for which not all Pij are diagonal. Due to noncommutativity, a complete set of common eigenfunctions of all Pij cannot exist. These states are basis functions of higher-dimensional representations of the permutation group. These states are not realized in nature; they are referred to 3 QM I, Chap. 16.
1.Second Quantization as parasymmetric states..The fictitious particles that are described by these states are known as paraparticles and are said to obey parastatis- tics. 1.1.2 Examples (i)Two 2)be an arbitrary wave function.The permutation Pa leads to P(1,2) From these two wave functions one can form (1.2)+(2.1)symmetric (1.112) =(1.2)-(2.1)antisvmmetric r the example of a wave function that is a function only of the spatial (1,2,3)=(E1,x2,x3) Application of the permutation Ps yields P乃28(r1,x2,xg）=r2,3,1) ie 1,23)=-92= cle 3 by particle 1. B2(1,2,3)=e We consider A3P21,2,3)=P3(2,1,3)=(2,3,1)=P23(1,2,3) 2P3(1,2,3)=2(3.2,1)=(3,1,2）)=P32(1,2,3) P123)1.2,3)=P123(2,3.1)=3.1.2)=321,2.3) Clearly,.BsA2≠P2P3. Ss,the permutation group for three objects,consists of the following 3!=6 ele- S={1,A2,Pa,乃1，P23,P2=(A2s)2 (1.1.13) We now consider the effect of a permutation P on a ket vector.Thus far we have only allowed P to act on spatial direct product lo）=e12233(r1,x2,x3) (1.1140 A.M.L.Messiah and O.W.Greenberg,Phys.Rev.B 136,248 (1964),B 138
6 1. Second Quantization as parasymmetric states.4. The fictitious particles that are described by these states are known as paraparticles and are said to obey parastatistics. 1.1.2 Examples (i) Two particles Let ψ(1, 2) be an arbitrary wave function. The permutation P12 leads to P12ψ(1, 2) = ψ(2, 1). From these two wave functions one can form ψs = ψ(1, 2) + ψ(2, 1) symmetric ψa = ψ(1, 2) − ψ(2, 1) antisymmetric (1.1.12) under the operation P12. For two particles, the symmetric and antisymmetric states exhaust all possibilities. (ii) Three particles We consider the example of a wave function that is a function only of the spatial coordinates ψ(1, 2, 3) = ψ(x1, x2, x3). Application of the permutation P123 yields P123 ψ(x1, x2, x3) = ψ(x2, x3, x1), i.e., particle 1 is replaced by particle 2, particle 2 by particle 3, and particle 3 by particle 1, e.g., ψ(1, 2, 3) = e−x2 1(x2 2−x2 3)2 , P12 ψ(1, 2, 3) = e−x2 2(x2 1−x2 3)2 , P123 ψ(1, 2, 3) = e−x2 2(x2 3−x2 1)2 . We consider P13P12 ψ(1, 2, 3) = P13 ψ(2, 1, 3) = ψ(2, 3, 1) = P123 ψ(1, 2, 3) P12P13 ψ(1, 2, 3) = P12 ψ(3, 2, 1) = ψ(3, 1, 2) = P132 ψ(1, 2, 3) (P123) 2ψ(1, 2, 3) = P123 ψ(2, 3, 1) = ψ(3, 1, 2) = P132 ψ(1, 2, 3). Clearly, P13P12 = P12P13 . S3, the permutation group for three objects, consists of the following 3! = 6 elements: S3 = {1, P12, P23, P31, P123, P132 = (P123) 2 }. (1.1.13) We now consider the effect of a permutation P on a ket vector. Thus far we have only allowed P to act on spatial wave functions or inside scalar products which lead to integrals over products of spatial wave functions. Let us assume that we have the state |ψ = X x1,x2,x3 direct product z }| { |x11 |x22 |x33 ψ(x1, x2, x3) (1.1.14) 4 A.M.L. Messiah and O.W. Greenberg, Phys. Rev. B 136, 248 (1964), B 138, 1155 (1965).
1.1 Identical Particles,Many-Particle States,and Permutation Symmetry 7 with t山(x1. 2,x3)=(1l1(2l2(l3〉.nz the particle is labeled by the num and the spatial c The effect of Ps,for example,is defined as follows Pi23)= 2r23r31(1,2,x3) =∑r31z1)2z23(r1,2,x3) x123 In the second line the basis vectors of the thre particles in the direct product ar once more written in the usual order,1,2.3.We can now rename the summation follob thaccording to ())Frotist Pi2sl)=>)a)23)s (2,3,). 1,2,3 If the state has the wave function),the n P)has the wave function P(a).The parti and apply the elements of the group S,we get the six states from the state lo)h la)13)b) n2lal闭hW=l3lah）,Psla)lh〉=la)h)l9, Pla)l闭h）=h〉l3la), (1.1.15) Pi2s la)118)2h)3=la)218)3h)1=h)la)18), 32la)l3)〉=3)la) t out etc.).It is the particles that are permutated,not the ar ruments of the states and y are all different,then the same is true of the six Invariant subspaces: Basis 1(symmetric basis): √后a)lh》+l1a)h)+ahwl间+hml间1a)+hmla)l画+1)e》) (1.1.16a Basis 2(antisymmetric basis): a)l间h)-l1ah)-a)hwl间-)+间+1) (1.1.16b) 5 An invariant subspace is a subspace of states which transforms into itself on application of the group elements
1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 7 with ψ(x1, x2, x3) = x1| 1 x2| 2 x3| 3|ψ. In |xij the particle is labeled by the number j and the spatial coordinate is xi. The effect of P123, for example, is defined as follows: P123 |ψ = X x1,x2,x3 |x12 |x23 |x31 ψ(x1, x2, x3) . = X x1,x2,x3 |x31 |x12 |x23 ψ(x1, x2, x3) In the second line the basis vectors of the three particles in the direct product are once more written in the usual order, 1,2,3. We can now rename the summation variables according to (x1, x2, x3) → P123(x1, x2, x3)=(x2, x3, x1). From this, it follows that P123 |ψ = X x1,x2,x3 |x11 |x22 |x33 ψ(x2, x3, x1) . If the state |ψ has the wave function ψ(x1, x2, x3), then P |ψ has the wave function P ψ(x1, x2, x3). The particles are exchanged under the permutation. Finally, we discuss the basis vectors for three particles: If we start from the state |α |β |γ and apply the elements of the group S3, we get the six states |α |β |γ P12 |α |β |γ = |β |α |γ , P23 |α |β |γ = |α |γ |β , P31 |α |β |γ = |γ |β |α , P123 |α1 |β2 |γ3 = |α2 |β3 |γ1 = |γ |α |β , P132 |α |β |γ = |β |γ |α . (1.1.15) Except in the fourth line, the indices for the particle number are not written out, but are determined by the position within the product (particle 1 is the first factor, etc.). It is the particles that are permutated, not the arguments of the states. If we assume that α, β, and γ are all different, then the same is true of the six states given in (1.1.15). One can group and combine these in the following way to yield invariant subspaces 5: Invariant subspaces: Basis 1 (symmetric basis): 1 √6 (|α |β |γ + |β |α |γ + |α |γ |β + |γ |β |α + |γ |α |β + |β |γ |α) (1.1.16a) Basis 2 (antisymmetric basis): 1 √6 (|α |β |γ−|β |α |γ−|α |γ |β−|γ |β |α + |γ |α |β + |β |γ |α) (1.1.16b) 5 An invariant subspace is a subspace of states which transforms into itself on application of the group elements
1.Second Quantization Basis 3: ∫a21ay1同h>+21网1@hW-la,hl间-hwla1ad -I)la)18)-18)y)la)) (1.1.16c) (0+0-la)h)13)+)13)la)+h)1a)13)-l3)hl1a) Basis 4: ((0+0-la))18)+1)18)la)-17)la)18)+13)h)la)) (2la)13)h)-21)la)h)+la)1a)h)1a)la) (1.1.16d -1)la)18)-18)h)la)) In the bases 3 and 4.the first of the t Pz and the second is odd under P(immediately below we shall call these twe functionsand).Other operations give rise to a linear combination of the two functions A2）=）,A2l2)=-2〉 (1.1.17a Psl)=a11）+a122〉，Psl2）=a21lp1）+a2l2〉， (1117b) with coefficients a.In matrix form,(1.1.17b)can be written as n(g)=（)(胸) (1.1.17c) The elements P and Pi3 are thus represented by 2x2 matrices m-(6-9)A-(aam） (1.1.18 This fact implies that the basis vectors and )span a two-dimensional repre 1.2 Completely Symmetric and Antisymmetric States We begin with the single-particle states i):1),2),....The single-particle states of the particles 1,2,...,a,....N are denoted by i:li)2,...,li ...i)N.These enable us to write the basis states of the N-particle system li1…,ia,…,iw）=lh…lia）a.liw)n (1.2.1) where particle 1 is in state li)and particle a in state li)etc.(The subscript outside the ket is the number labeling the particle,and the index within the ket identifies the state of this particle.) Provided that the {i)}form a complete orthonormal set,the product states defined above likewise represent a complete orthonormal system in the
8 1. Second Quantization Basis 3: 8 < : √1 12 (2 |α |β |γ + 2 |β |α |γ−|α |γ |β−|γ |β |α − |γ |α |β−|β |γ |α) 1 2 (0 + 0 − |α |γ |β + |γ |β |α + |γ |α |β−|β |γ |α) (1.1.16c) Basis 4: 8 < : 1 2 (0 + 0 − |α |γ |β + |γ |β |α−|γ |α |β + |β |γ |α) √1 12 (2 |α |β |γ − 2 |β |α |γ + |α |γ |β + |γ |β |α − |γ |α |β−|β |γ |α) . (1.1.16d) In the bases 3 and 4, the first of the two functions in each case is even under P12 and the second is odd under P12 (immediately below we shall call these two functions |ψ1 and |ψ2). Other operations give rise to a linear combination of the two functions: P12 |ψ1 = |ψ1 , P12 |ψ2 = − |ψ2 , (1.1.17a) P13 |ψ1 = α11 |ψ1 + α12 |ψ2 , P13 |ψ2 = α21 |ψ1 + α22 |ψ2 , (1.1.17b) with coefficients αij . In matrix form, (1.1.17b) can be written as P13 „ |ψ1 |ψ2 « = „ α11 α12 α21 α22 « „ |ψ1 |ψ2 « . (1.1.17c) The elements P12 and P13 are thus represented by 2 × 2 matrices P12 = „ 1 0 0 −1 « , P13 = „ α11 α12 α21 α22 « . (1.1.18) This fact implies that the basis vectors |ψ1 and |ψ2 span a two-dimensional representation of the permutation group S3. The explicit calculation will be carried out in Problem 1.2. 1.2 Completely Symmetric and Antisymmetric States We begin with the single-particle states |i: |1, |2, ... . The single-particle states of the particles 1, 2, ... , α, ... , N are denoted by |i1, |i2, ... , |iα, ... , |iN . These enable us to write the basis states of the N-particle system |i1,... ,iα,... ,iN = |i11 ... |iαα ... |iN N , (1.2.1) where particle 1 is in state |i11 and particle α in state |iαα, etc. (The subscript outside the ket is the number labeling the particle, and the index within the ket identifies the state of this particle.) Provided that the {|i} form a complete orthonormal set, the product states defined above likewise represent a complete orthonormal system in the