The general matrix elements of the Boltzmann operator may now be expressed using the Euclidean
path integral,
〈q′|e−βH|q〉=∫
Dqexp{
−1
̄h∫ ̄hβ0dτ LE(q,q ̇)}
(15.45)subject to the boundary conditions,
q(0) = q
q( ̄hβ) = q′ (15.46)Here, LE is the Euclidean Lagrangian corresponding to the HamiltonianH. To complete the
construction of the partition function, we need to setq′=qand then integrate over allq. But this
instruction simply means that we should integrate in the path integral over all functionsq(τ) which
are periodic inτwith period ̄hβ. Thus, we arrive at the following final formula for the partition
function,
Z=∫
Dqexp{
−1
̄h∫ ̄hβ0dτ LE(q,q ̇)}
q( ̄hβ) =q(0) (15.47)and it is understood that the actual value ofq( ̄hβ) =q(0) is also integrated over, as part of the
functional integral
∫
Dq.15.6 Classical Statistical Mechanics as the high temperature limit
In the largeT (equivalently, smallβ) limit, many quantum states are occupied, and the system is
expected to behave classically. It is possible to recover this limit from the general path integral
formula given above. We carry out the limit here as an illustration of path integral methods.
In the limitβ→0, the interval of periodicity ofq(τ) becomes small. It is useful to decompose
q(τ) in an orthonormal basis of periodic functions on the interval [0, ̄hβ],
q(τ) =+∑∞n=−∞cne^2 πinτ/( ̄hβ) c∗n=c−n (15.48)Thereality conditionc∗n=c−nis needed to guarantee that the functionsq(τ) be real. Thethermal
orMatsubara frequencies
ωn=
2 πn
̄hβ(15.49)
govern the magnitude of the kinetic terms inLE. Whenβ→0, the frequenciesωnbecome large as
long asn 6 = 0. The frequencyω 0 vanishes for allβand requires zero cost in kinetic energy for this
mode. Thus, the kinetic energy will suppress all the modes withn 6 = 0, and only the moden= 0
will survive at high temperature. We shall denote this mode byc 0 =q. As a result,
Z(β→0) =∫+∞−∞dq e−βV(q)∫
D′qexp{
−1
̄h∫ ̄hβ0dτ1
2
mq ̇^2}
(15.50)