1549380323-Statistical Mechanics Theory and Molecular Simulation

Quantum equilibrium ensembles 401

can exploit the quantum-classical correspondence principle and simply promote the
classical equilibrium phase space distribution functions, all of which are functions of
the classical Hamiltonian, to quantum operators. Thus, for the canonical ensemble at
temperatureT, the normalized density operator becomes

ρ ̃(Hˆ) =

e−βHˆ

Q(N,V,T)

. (10.4.2)

Since ̃ρmust have unit trace, the partition function is given by

Q(N,V,T) = Tr

[

e−β

Hˆ]

. (10.4.3)

Here,Q(N,V,T) is identified with the numberZ, the total number of microscopic
states in the ensemble. Thus, the unnormalized density matrix ˆρis exp(−βHˆ). Casting
eqns. (10.4.2) and (10.4.3) into the basis of the eigenvectors ofHˆ, we obtain

〈Ek|ρ ̃|Ek〉=

e−βEk

Q(N,V,T)

Q(N,V,T) =

∑

k

e−βEk. (10.4.4)

Eqn. (10.4.4) indicates that the microscopic states correspondingto the canonical
ensemble are eigenstates ofHˆ, and the probability of any member of the ensemble being
in a state|Ek〉is exp(−βEk)/Q(N,V,T). OnceQ(N,V,T) is known from eqn. (10.4.4),
the thermodynamics of the canonical ensemble are determined as usual from eqn.
(4.3.23). Finally, the expectation value of any operatorAˆin the canonical ensemble is
given by

〈Aˆ〉= Tr

(

ρ ̃Aˆ

)

=

1

Q(N,V,T)

∑

k

e−βEk〈Ek|Aˆ|Ek〉. (10.4.5)

(Feynman regarded eqns. (10.4.4) and (10.4.5) as the core of statistical mechanics, and
they appear on the first page of his bookStatistical Mechanics: A Set of Lectures.)^2 )
If there are degeneracies among the eigenvalues, then a factorg(Ek), which is the
degeneracy of the energy levelEk, i.e., the number of independent eigenstates with this
energy, must be introduced into the above sums over eigenstates. Thus, for example,
the partition function becomes

Q(N,V,T) =

∑

k

g(Ek)e−βEk, (10.4.6)

and the expectation value of the operatorAˆis given by

(^2) In reference to eqn.(10.4.5), Feynman casually remarks, “This law is the summit of statistical
mechanics, and the entire subject is either the slide-down from this summit, as the principle is
applied to various cases, or the climb-up to where the fundamental law is derived and the concepts
of thermal equilibrium and temperatureTclarified” (Feynman, 1998). Our program in the next two
chapters will be the former, as we apply the principle and develop analytical and computational tools
for carrying out quantum statistical mechanical calculations for complex systems.