where 1/;t is the Hermitian conjugate of 1/; and [A, B] == AB - BA, {A, B} == AB + BA.
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SPECIAL TOPICS IN STATISTICAL MECHANICS
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The definition of the quantized field is completed by defining two Hermitian operators-the Hamiltonian operator .Yt' and the number operator Nop . The Hamiltonian operator is
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.Yt'=K+f2 h2 K = 2m
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(A.44)
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where u 12
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r1 d3r2I/;t(rl)l/;t(r2)U12I/;(r2)1/;(rl)
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= u(r
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The number operator is (A.45)
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These definitions hold for both bosons and fermions. We can easily verify that (A.46) Therefore .Yt' and Nop can be simultaneously diagonalized. We show that a simultaneous eigenstate of .Yt' and Nop is an energy eigenstate of a system of a definite number of particles. Let a complete orthonormal basis of the Hilbert space be so chosen that any vector I<Pn) of the basis is a simultaneous eigenstate of .Yt' and Nop . Let a particular member of the basis be denoted by I'!'EN)' with the properties that
('!'ENI'!'EN) I'!'EN)
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The state 10) properties are
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EI'!'EN)
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(A.47)
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Nopl '!'EN) = NI '!'EN)
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= I'!'oo), called the vacuum state, is assumed to be unique. Its
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10) NoplO)
-I/;t(r)
(A.48)
From (A.45) and (A.43) it is easily verified that
[I/;(r), Nop ] = I/;(r) [I/;t(r), Nop ]
Hence
(A.49)
Nopl/;(r)I'!'EN) = (N - l)l/;(r)I'!'EN)
Nopl/;t(r)I'!'EN)
(N + l)1/;t(r)I'!'EN)
(A.50)
Thus I/;(r) decreases N by 1, and I/;t(r) increases N by 1. By repeated application of I/;t(r) to 10), we prove that the eigenvalues of Nop are
N=0,1,2, ...
(A.51)
N-BODY SYSTEM OF IDENTICAL PARTICLES
Since I/;(r) decreases N by 1, and the state with N we have the identity
is assumed to be unique, (A.52)
where I/;(j) == I/;(r). Let a function of the N position coordinates r l , ... , r N be defined by (A.53) By (A.43) this function is symmetric (antisymmetric) with respect to the exchange of any two coordinates for bosons (fermions). The norm of 'l'EN(l, ... , N) is unity, i.e., (A.54)
Proof
1d 3Nr 'l'tN(l, ... , N)'l'EN(l, ... , N) 1 = N! 1 3Nr('l'ENll/;t(N) ... I/;t(l)IO)(OII/;(l) ... I/;(N)I'l') d
By (A.52) we can write
-1 d 3Nr I: ('l'ENll/;t(N) ... I/;t(l)llP )(lP ll/;(l) .. , I/;(N)I'l'EN) N!
= N!
1d 3Nr('l'ENI[l/;t(N) ... I/;t(l)][I/;(l) .. I/;(N)]I'l'EN)
Now carry out the integration over rl' The relevant factor is
Next carry out the integration over r 2 . The relevant factor is
By induction we can show that
The connection between the quantized field and an N-body system is furnished by the following theorem.
THEOREM
SPECIAL TOPICS IN STATISTICAL MECHANICS
(A.55)
Proof
By (A.47) and (A53)
v'NT (01 [HI) ... o/(N)] 1 'YEN> = E'YEN (I, ... , N)
(A.56)
Since 1 0> = 0, and is Hermitian, we also have (01 = O. Hence the left side of (A56) has the form of a commutator:
v'NT(01[0/(1) ... o/(N)] 1'YEN >
= v'NT (01[0/(1) HN), ] 1 'YEN> = v'NT j~l (010/(1) [o/(J), ] ... o/(N)I'YEN )
where the last step is obtained through repeated use of the identity
[AB, C]
(A.57)
[A, C]B
+ A[B, C]
We explicitly calculate [o/(J), ). From (A44) we have
[o/(i), ]
For Bosons
[o/(J), K] + [o/(J), g]
[o/(J), K] = - 2m
1d 3r [o/(J), o/t(r)V' 20/(r)] h = - - 1 r [0/ (J ), 0/t(r)] V' 20/ (r) d 2m
= - - V'20/(J) 2m J [o/(J), g] = ~ d 3 1 d 3 2 [o/(J), o/t(1)o/t(2)] v12 0/(2) 0/(1) r r
1 = ~ 1 3r d3r2 {[ 0/ (J), o/t(l)] o/t(2) + o/t(l) [0/ (J), o/t(2)]} d
Xv 12 0/(2)0/(1)
[1 d 3ro/ (r)v(r,r )o/(r)]0/(J)
N-BODY SYSTEM OF IDENTICAL PARTICLES
For Fermions
liZ [0/ (J), K]
2m 2m
f d 3 [0/ (J), o/t(r) V zo/(r)] r
liz = - - f d 3 {o/(J), o/t(r)} vZo/(r) r liz
VZo/(J)
[Hi), f.l] = }f d 3r1 d 3rz [o/(J), o/t(1)o/t(2)]u 12 0/(2)o/(1)
f d 3r1 d3rz [{0/ (J), o/t(l)} o/t(2) - o/t(l){ 0/ (J), o/t(2)} ]
Xu 12 0/(2)0/(1) = [f d 3 o/t(r) u(r, rj)o/(r)] o/(J) r
Hence for both bosons and fermions we have
where
X(J) = f d 3ro/ t (r)u(r,r j )0/(r)
The following properties of X(j) are trivial:
(A.59)
[o/(i), X(J)] = uijo/(i) X(J)IO) = 0, (OIX(J) =
Substitution of (A59) into (AS ) yields
(A.60) (A.61)
liz N = - L V/'I'EN(l, ... , N) + {NT j~l ( 10/(1) ... o/(J - l)X(J)o/(J) ... o/(N)I'I'EN) (A.62)
2m j~l 1 N
We now commute X(j) all the way to the left with the help of (A60):
[0/(1) o/(J - l)X(J)o/(J) o/(N)] = [0/(1) o/(J - 2)X(J)o/(J - 1) o/(N)] +uj _l,j[o/(l) ... o/(N)] = [0/(1)'" o/(J - 3)X(J)0/(J - 2) o/(N)] +(uj-Z,j + Uj - 1 ,j)[0/(1). . o/(N)]
[X(J) +
~~>ij][o/(l) ... o/(N)]
(A.63)
SPECIAL TOPICS IN STATISTICAL MECHANICS
Substituting this into (A.62) and using (A.61) we obtain
L \7/+ LVi)
j=l i<j
'YEN(l, ... ,N)
(A.64)
For convenience in actual calculations, let us introduce a complete orthonormal set of single-particle wave functions {u,,(r)} with
Then we may expand the field operators I/;{r) and I/;t(r) in the following manner:
I/;(r) =
La"u,,(r)
(A.65)
where, in accordance with (A.43), a" and a! are operators satisfying the following commutation rules:
Bosons Fermions
(A.66)
It easily follows that the eigenvalues of a!a" are
(bosons) (fermions) In terms of a" and a! we have
(A.67)
(A.68)
where
(0:1 - \7 2 1,8)
f d ru:\72 uf3
(o:,8l v lyA)
r1 d 3 r2 u:(1)u;(2)v 12 uy(1)u>.(2)
(A.70)
Let a set of integers {no, n 1, } be given, such that each n" is a possible value of a!a" as given by the rule (A.67). Define the state In) by
In) == Ino, nl> ... ) ==
