QuantumPhysics.dvi

Theinternal energyEand theentropySare related to the free energy by

F=E−TS (15.37)

Both may be expressed in terms of the partition function as well,

E = −

∂lnZ

∂β

S = −

∂F

∂T

(15.38)

It is instructive to work out the internal energy in a basis ofenergy eigenstates,

E=

1

Z

∑

n

Ene−βEn (15.39)

A useful interpretation of these expressions is obtained byintroducing the notion ofthe statistical
probabilityPnfor finding the system in a statenwith energyEn. The statistical probability is
defined as the normalized Boltzmann weight,

Pn≡

e−βEn

Z

∑

n

Pn= 1 (15.40)

The internal energy then takes on a natural form,

E=

∑

n

EnPn (15.41)

The entropy may also be conveniently expressed in terms of thePn,

S=−kB

∑

n

PnlnPn (15.42)

Some authors (such as Feynman) actually take the last formula as the definition for the entropy.
This definition is also closely related to information theory definition of entropy (see Shannon).

15.5 Path integral formulation of quantum statistical mechanics

There is a simple and suggestive formula for the partition function (and in view of the preceding
subsection, thus for all thermodynamic quantities) in terms of the Euclidean path integral. The cor-
respondence between the Boltzmann operatore−βHand the Euclidean evolution operatore−τβH/ ̄h
is fully brought out by the equality

̄hβ=τβ (15.43)

The trace, needed for the partition function, may be expressed as an integral overq,

Z= Tre−βH=

∫

dq〈q|e−βH|q〉 (15.44)