QuantumPhysics.dvi

For the free case (V = 0), we haveLE=^12 mq ̇^2 , so that

〈qb|UE(τb−τa)|qa〉=

√

m

2 π ̄h(τb−τa)

exp

{

−

m(qb−qa)^2

2 ̄h(τb−τa)

}

(15.30)

which is truly a Gaussian. Notice that for the free particle,this matrix element satisfies the heat
equation,
(
∂
∂τ

+

̄h

2 m

∂^2

∂q^2

)

〈q|UE(τ−τa)|qa〉= 0 (15.31)

with the initial condition that

〈q|UE(τa−τa)|qa〉=δ(q−qa) (15.32)

Its physical interpretation is as follows. We place aδ-function heat distribution at the pointqa
at timeta, and then observe the heat diffuse overqas a function of timeτ; this is given by the
function〈q|UE(τ−τa)|qa〉.

15.4 Quantum Statistical Mechanics

The operatore−τH/ ̄his closely related to the Boltzmann operatore−βHof statistical mechanics.
We introduce the standard notation,

β≡

1

kBT

(15.33)

whereT is temperature andkBis the Boltzmann constant. In any energy eigenstate|En〉ofH,
the Boltzmann operator takes on the definite value

e−βH|En〉=e−βEn|En〉 (15.34)

and thus provides the standard Boltzmann weight for a state with energyEn.

The starting point of equilibrium statistical mechanics inthe canonical ensemble is thepartition
function, which is defined by

Z≡Tr

(

e−βH

)

=

∑

n

e−βEn (15.35)

In the second equality, we have expanded the trace in a basis of eigenstates ofHwhich are labeled
byn, and whose energy isEn. Their multiplicities are properly included by summing, for eachEn
over all the states with energyEn. All other thermodynamic functions may be expressed in terms
of the partition function. Thefree energyFis defined by

Z=e−βF or F=−

1

β

lnZ=−kBTlnZ (15.36)