1549380323-Statistical Mechanics Theory and Molecular Simulation

402 Quantum ensembles

〈Aˆ〉=

1

Q(N,V,T)

∑

k

g(Ek)e−βEk〈Ek|Aˆ|Ek〉. (10.4.7)

In an isothermal-isobaric ensemble at temperatureTand pressureP, the density
operator, partition function, and expectation value are given, respectively, by

ρ ̃(Hˆ,V) =

e−β(Hˆ+PV)

∆(N,P,T)

,

〈Ek| ̃ρ(Hˆ,V)|Ek〉=

e−β(Ek+PV)

∆(N,P,T)

, (10.4.8)

∆(N,P,T) =

∫∞

0

dVTr

[

e−β(

Hˆ+PV)]

=

∫∞

0

dV

∑

k

e−β(Ek+PV), (10.4.9)

〈Aˆ〉=

1

∆(N,P,T)

∫∞

0

dVTr

[

Aˆe−β(Hˆ+PV)

]

=

1

∆(N,P,T)

∫∞

0

dV

∑

k

e−β(Ek+PV)〈Ek|Aˆ|Ek〉. (10.4.10)

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

∆(N,P,T) =

∫∞

0

dV

∑

k

g(Ek)e−β(Ek+PV)

〈Aˆ〉=

1

∆(N,P,T)

∫∞

0

dV

∑

k

g(Ek)e−β(Ek+PV)〈Ek|Aˆ|Ek〉. (10.4.11)

Finally, for the grand canonical ensemble at temperatureTand chemical potential
μ, the density operator, partition function, and expectation valueare given by

ρ ̃(Hˆ,N) =

e−β(Hˆ−μN)

Z(μ,V,T)

,

〈Ek|ρ ̃(Hˆ,N)|Ek〉=

e−β(Ek−μN)

Z(μ,V,T)

, (10.4.12)

Z(μ,V,T) =

∑∞

N=0

Tr

[

e−β(

Hˆ−μN)]