Statistical Mechanics - Physics at Oregon State University

6.2. GENERAL ENSEMBLES. 123
We can now make the following statement. In general, an arbitrary system is
described by a density operator ρ, which is any operator satisfying the conditions
(6.4) through (6.6). We also use the term density matrix, but this is basis
dependent. If we have an arbitrary basis |n〉, the density matrix R corresponding
to ρ in that basis is defined by Rij = 〈i| ρ |j〉. In statistical mechanics we
make a particular choice for the density matrix. For example, a system at a
temperature T and volume V , number of particles N has the density operator
given by (6.12). The extensive state variables N and V are used in the definition
of the Hamiltonian.
6.2 General ensembles.
The partition function is related to the Helmholtz free energy by Z = e−βF .
The states |n〉 in the definitions above all have N particles, and the density
operator ρ is defined in the Hilbert space of N-particle states, with volume V. It
is also possible to calculate the grand potential Ω in a similar way. The number
operator N gives the number of particles of state |n〉 via N |n〉 = Nn |n〉. The
energy of this state follows from H |n〉 = En |n〉.
In this space we still have orthogonality and completeness of the basis. The
only difference with the example in the previous section is that our Hilbert space
is much larger. We now include all states with arbitrary numbers of particles
(even zero!!), but still with volume V. Mathematically these spaces are related
by SV =
N SV,N .
The grand partition function Z is defined by
Z =
e −β(En−µNn)
(6.19)
n
where we sum over states |n〉 with arbitrary numbers of particles. This is often
desirable in field-theory, where the number of particles is not fixed and the
operators are specified in terms of creation and annihilation operators on this
larger space, called Fock-space. The grand partition function is a trace over all
states in this general Fock-space:
Z = Tr e −β(H−µN) = e −βΩ
(6.20)
and the density matrix in this case is 1
Ze−β(H−µN) . The operator H−µN is sometimes
called the grand Hamiltonian. In this derivation we have used the fact
that we can define energy eigenstates for specific particle numbers, or [H, N] = 0.
That condition is not necessary, however. We can use the partition function in
the form (6.20) even if the Hamiltonian does not commute with the particle
number operator, like in the theory of superconductivity. The density operator
is used to calculate probabilities. The probability of finding a system with
temperature T , chemical potential µ, and volume V in a state |x〉 is given by
〈x| 1
Ze−β(H−µN) |x〉.
It is also possible to use other Legendre transformations. If V is the volume
operator, one can show that