1549380323-Statistical Mechanics Theory and Molecular Simulation

Quantum equilibrium ensembles 403

=

∑∞

N=0

∑

k

e−β(Ek−μN), (10.4.13)

〈Aˆ〉=

1

Z(μ,V,T)

∑∞

N=0

Tr

[

Aˆe−β(Hˆ−μN)

]

=

1

Z(μ,V,T)

∑∞

N=0

∑

k

e−β(Ek−μN)〈Ek|Aˆ|Ek〉. (10.4.14)

As before, if there are degeneracies, then a factor ofg(Ek) must be introduced into
the above sums:

Z(μ,V,T) =

∑∞

N=0

∑

k

g(Ek)e−β(Ek−μN)

〈Aˆ〉=

1

Z(μ,V,T)

∑∞

N=0

∑

k

g(Ek)e−β(Ek−μN)〈Ek|Aˆ|Ek〉. (10.4.15)

The quantum grand canonical ensemble will prove particularly useful in our treatment
of the quantum ideal gases to be discussed in Chapter 11.
In the above list of definitions, a definition of the quantum microcanonical ensemble
is conspicuously missing for the reason that it is very rarely used forcondensed-phase
systems. Moreover, in order to define this ensemble, the quantum-classical correspon-
dence must be applied carefully because the eigenvalues ofHˆ are assumed to be dis-
crete. Hence, theδ-function used in the classical microcanonical ensemble does not
make sense for quantum systems because a given eigenvalue may ormay not be equal
to the energyEused to define the ensemble. However, if we define an energy shell
betweenEandE+ ∆E, then we can certainly find a subset of energy eigenvalues in
this shell. The partition function is then related to the number of energy levelsEk
satisfyingE < Ek< E+ ∆E. Typically, when we take the thermodynamic limit of a
system, the energy levels become very closely spaced, and we can shrink the thickness
∆Eof the shell to zero.

10.4.1 Example: The canonical harmonic oscillator

In order to illustrate the application of a quantum equilibrium ensemble, we consider
the case of a simple one-dimensional harmonic oscillator of frequencyω. We will derive
the properties of this system using the canonical ensemble. Recallfrom Section 9.3 that
the energy eigenvalues are given by

En=

(

n+

1

2

)

̄hω n= 0, 1 , 2 ,.... (10.4.16)

The canonical partition function is, therefore,