QuantumPhysics.dvi

16.5 Quantum Statistical Mechanics

One particularly important special case of mixed states andensembles is provided by equilibrium
quantum statistical mechanics. Consider a system described by a time independent Hamiltonian
H, whose eigenstates and eigenvalues will be denoted by|En;α〉andEnrespectively. Hereαstands
for additional quantum numbers which characterize differentstates at the same energyEn. The
states|En;α〉will be assumed to be orthonormal. Already in Chapter 8 have we introduced the
standard statistical mechanical quantities such as the partition functionZ, and the Boltzmann
weightpnof a state with energyEn, given by

Z = Tr

(

e−βH

)

β=

1

kBT

pn =

e−βEn

Z

(16.22)

From the Boltzmann weights, we can now compute the density operator,

ρ=

∑

n,α

|En;α〉pn〈En;α| (16.23)

Using the fact that in the orthonormal basis|En;α〉we have,

∑

n,α

|En;α〉e−βEn〈En;α|=e−βH (16.24)

the density operator takes on a particularly simple form,

ρ=

e−βH

Tr (e−βH)

(16.25)

This expression makes the defining properties of the densityoperator manifest; it is self-adjoint,
sinceH is; it has unit trace; and it is non-negative sinceH is self-adjoint and has only real
eigenvalues. Other thermodynamic quantities similarly have simple expressions. The internal
energy is the ensemble average of the Hamiltonian,

E=

1

Z

∑

n,α

Ene−βEn= Tr(ρE) (16.26)

Similarly, the entropy is given by

S=−kB

∑

n,α

pnlnpn=−kBTr

(

ρlnρ

)

(16.27)

In fact, one may take the entropy as the starting point for thermodynamics, and then use it to
derive the Boltzmann distribution. Working in the canonical ensemble, one keeps the energy fixed
and maximizes the entropy given this constraint. The standard way of doing this in practice is to