458 The Feynman path integral
common case for which we need to evaluate eqn. (12.3.2) is ultimately the simplest. If
Aˆis purely a function of ˆx, then|x〉is an eigenvector ofAˆ(ˆx) satisfying
Aˆ(ˆx)|x〉=a(x)|x〉, (12.3.3)wherea(x) is the corresponding eigenvalue, and eqn. (12.3.2) reduces to
〈Aˆ〉=
1
Q(L,T)
∫
dx a(x)〈x|e−β
Hˆ
|x〉. (12.3.4)Thus, for operators that are functions only of position, eqn. (12.3.4) indicates that only
the diagonal elements of the density matrix are needed. Substituting eqn. (12.2.22) for
x=x′into eqn. (12.3.4) leads to a path integral expression for the expectation value
ofAˆ(ˆx):
〈Aˆ〉=
1
Q(L,T)
lim
P→∞(
mP
2 πβ ̄h^2)P/ 2 ∫
dx 1 ···dxPa(x 1 )×exp{
−
1
̄h∑P
k=1[
mP
2 β ̄h(xk+1−xk)^2 +β ̄h
PU(xk)]}∣∣
∣
∣
∣
xP+1=x 1. (12.3.5)
Although eqn. (12.3.5) is perfectly correct, it appears to favor one particular posi-
tion variable (x 1 ) over the others, sincea(x) is evaluated only at this point. Eqn.
(12.3.5) will consequently converge slowly and is not particularly useful for actual
computations. Because the paths are cyclic, however, all pointsx 1 ,...,xP of a path
are equivalent. The equivalence can be proved by noting that the argument of the
exponential is invariant under a cyclic relabeling of the coordinate variables
x′ 2 =x 1 , x′ 3 =x 2 , ···x′P=xP− 1 , x′ 1 =xP. (12.3.6)If such a relabeling is introduced into eqn. (12.3.5), a completely equivalent expression
for the expectation value results:
〈Aˆ〉=
1
Q(L,T)
lim
P→∞(
mP
2 πβ ̄h^2)P/ 2 ∫
dx′ 1 ···dx′Pa(x′ 2 )×exp{
−
1
̄h∑P
k=1[
mP
2 β ̄h(x′k+1−x′k)^2 +β ̄h
PU(x′k)]}∣∣
∣
∣
∣
x′P+1=x′ 1. (12.3.7)
A second relabeling,x′′ 3 =x′ 2 ,x′′ 4 =x′ 3 ,....,x′′P=x′P− 1 ,x′′ 1 =x′P,x′′ 2 =x′ 1 , would yield
a similar expression witha(x) evaluated atx′′ 3. SincePsuch relabelings are possible,
we can derivePequivalent expressions for the expectation value, each involving the
evaluation of thea(x) at the different coordinatesx 1 ,...,xP. If these expressions are
added together and divided byP, we find