QuantumPhysics.dvi

The contribution of the second factor is being kept as a subdominant effect, which may be evaluated
since it corresponds to the partition function of the free particle. Here,D′qstands for the instruction
that the constant mode corresponding toc 0 has to be removed from this integration. In terms of
the Fourrier component modes, this measure is given by

D′q=

∏

n 6 =0

dcn (15.51)

We then have

∫

D′qexp

{

−

1

̄h

∫ ̄hβ

0

dτ

1

2

mq ̇^2

}

=

√

m

2 π ̄h^2 β

(15.52)

Comparing with the partition function of classical statistical mechanics,

Zclassical=

∫ ∫

dpdq e−βH(q,p)=

∫+∞

−∞

dq e−βV(q)

∫+∞

−∞

dpe−βp

(^2) /(2m)
(15.53)
we find that
Z(β→0) =

1

√

2 π^2 ̄h^2

Zclassical (15.54)

The constant factor merely reflects an unphysical overall normalization. It implies that the classical
free energy differs from the quantum one by a constant shift, which is physically unobservable.