1549380323-Statistical Mechanics Theory and Molecular Simulation

472 The Feynman path integral

arex 1 andx 2 , then, as we saw in Section 9.4, the coordinate eigenvectors for bosons
and fermions take the form

1

√

2

[|x 1 x 2 〉+|x 2 x 1 〉] (bosons)

1

√

2

[|x 1 x 2 〉−|x 2 x 1 〉] (fermions),

respectively. Thus, for bosons, the partition function is given by

Q=

1

2

∫

dx 1 dx 2 [〈x 1 x 2 |+〈x 2 x 1 |] e−β

Hˆ

[|x 1 x 2 〉+|x 2 x 1 〉]

=

∫

dx 1 dx 2

[

〈x 1 x 2 |e−β

Hˆ

|x 1 x 2 〉+〈x 1 x 2 |e−β

Hˆ

|x 2 x 1 〉

]

, (12.5.1)

while for fermions, it is

Q=

∫

dx 1 dx 2

[

〈x 1 x 2 |e−β

Hˆ

|x 1 x 2 〉−〈x 1 x 2 |e−β

Hˆ

|x 2 x 1 〉

]

. (12.5.2)

Functional integral expressions for each of the two matrix elements appearing in eqns.
(12.5.1) and (12.5.2) can be derived using the techniques already developed in Sec-
tion 12.4. These expressions are

〈x 1 x 2 |e−β

Hˆ

|x 1 x 2 〉=

∫x 1 (β ̄h)=x 1 ,x 2 (β ̄h)=x 2

x 1 (0)=x 1 ,x 2 (0)=x 2

Dx 1 Dx 2 e−S[x^1 ,x^2 ]/ ̄h

〈x 1 x 2 |e−β

Hˆ

|x 1 x 2 〉=

∫x 1 (β ̄h)=x 2 ,x 2 (β ̄h)=x 1

x 1 (0)=x 1 ,x 2 (0)=x 2

Dx 1 Dx 2 e−S[x^1 ,x^2 ]/ ̄h. (12.5.3)

These two terms are illustrated in Fig. 12.9. In particular, note thatthe first term
involves two independent closed paths for particles 1 and 2, respectively, in which the
pathsx 1 (τ) andx 2 (τ) satisfyx 1 (0) =x 1 (β ̄h) =x 1 andx 2 (0) =x 2 (β ̄h) =x 2. This is
exactly the term that would result if the physical state of the system had no particular
symmetry and could be simply described as|x 1 x 2 〉or|x 2 x 1 〉. The second term, which
results from the symmetry conditions placed on the state vector,“ties” the paths
together at the endpoints because of the endpoint conditionsx 1 (0) =x 2 (β ̄h) =x 1 and
x 2 (0) =x 1 (β ̄h) =x 2. This second term, called anexchange term, is a purely quantum
mechanical effect arising from the symmetry of the state vector.Exchange effects
involve long-range correlations of delocalized wave functions and can often be neglected
for particles such as protons unless the system is at a very low temperature. For
electrons, however, such effects are nearly always important andneed to be included.
In order to underscore the difficulties associated with exchange effects, consider
writing eqns. (12.5.1) and (12.5.2) as limits of discrete path integrals.For compactness,

we use the notation of eqn. (12.2.28), with the discretized paths denoted asx(1) 1 ,...,x( 1 P)