Statistical Mechanics - Physics at Oregon State University

124 CHAPTER 6. DENSITY MATRIX FORMALISM.
Tr e −β(H+pV) = e −βG
(6.21)
where G is the Gibbs free energy G = U − T S + pV . The trace is a sum
over quantum states with arbitrary volumes. This is not necessarily a useful
extension, though. Quantum states are often easier to describe at constant
volume.
In summary, we have
Z(T, V, N) = Tr
SV,N
e −βH
(6.22)
where the trace is in the space of all quantum states with volume V and number
of particles N. Similarly, we have
Z(T, µ, N) = Tr
SV
e −β(H−µN)
(6.23)
where the trace is now in the much larger Hilbert space of states with volume V.
States do not have to have well defined particle numbers! Independent whether
the Hamiltonian commutes with the particle number operator we can always
write
or
Z(T, µ, N) =
Z(T, µ, N) =
Tr e −β(H−µN)
N
N
e βµN Tr
SV,N
SV,N
(6.24)
e −βH =
e βµN Z(T, V, N) (6.25)
as before. This can also be extended. Suppose we can measure the magnetic
moment of a system. This magnetic moment M will be an integer times a basic
unit for finite particle numbers. Hence we have
Z(T, V, N, M) = Tr
SV,N,M
N
e −βH
(6.26)
where the summation is over a smaller space with a specified value of the magnetic
moment. If we do calculations as a function of magnetic field we have:
We find again:
Z(T, V, N, h) = Tr
SV,N
e −β(H−hM)
(6.27)
Z(T, V, N, h) =
e βhM Z(T, V, N, M) (6.28)
M