Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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6.2. GENERAL ENSEMBLES. 123<br />
We can now make the following st<strong>at</strong>ement. In general, an arbitrary system is<br />
described by a density oper<strong>at</strong>or ρ, which is any oper<strong>at</strong>or s<strong>at</strong>isfying the conditions<br />
(6.4) through (6.6). We also use the term density m<strong>at</strong>rix, but this is basis<br />
dependent. If we have an arbitrary basis |n〉, the density m<strong>at</strong>rix R corresponding<br />
to ρ in th<strong>at</strong> basis is defined by Rij = 〈i| ρ |j〉. In st<strong>at</strong>istical mechanics we<br />
make a particular choice for the density m<strong>at</strong>rix. For example, a system <strong>at</strong> a<br />
temper<strong>at</strong>ure T and volume V , number of particles N has the density oper<strong>at</strong>or<br />
given by (6.12). The extensive st<strong>at</strong>e variables N and V are used in the definition<br />
of the Hamiltonian.<br />
6.2 General ensembles.<br />
The partition function is rel<strong>at</strong>ed to the Helmholtz free energy by Z = e−βF .<br />
The st<strong>at</strong>es |n〉 in the definitions above all have N particles, and the density<br />
oper<strong>at</strong>or ρ is defined in the Hilbert space of N-particle st<strong>at</strong>es, with volume V. It<br />
is also possible to calcul<strong>at</strong>e the grand potential Ω in a similar way. The number<br />
oper<strong>at</strong>or N gives the number of particles of st<strong>at</strong>e |n〉 via N |n〉 = Nn |n〉. The<br />
energy of this st<strong>at</strong>e follows from H |n〉 = En |n〉.<br />
In this space we still have orthogonality and completeness of the basis. The<br />
only difference with the example in the previous section is th<strong>at</strong> our Hilbert space<br />
is much larger. We now include all st<strong>at</strong>es with arbitrary numbers of particles<br />
(even zero!!), but still with volume V. M<strong>at</strong>hem<strong>at</strong>ically these spaces are rel<strong>at</strong>ed<br />
by SV = <br />
N SV,N .<br />
The grand partition function Z is defined by<br />
Z = <br />
e −β(En−µNn)<br />
(6.19)<br />
n<br />
where we sum over st<strong>at</strong>es |n〉 with arbitrary numbers of particles. This is often<br />
desirable in field-theory, where the number of particles is not fixed and the<br />
oper<strong>at</strong>ors are specified in terms of cre<strong>at</strong>ion and annihil<strong>at</strong>ion oper<strong>at</strong>ors on this<br />
larger space, called Fock-space. The grand partition function is a trace over all<br />
st<strong>at</strong>es in this general Fock-space:<br />
Z = Tr e −β(H−µN) = e −βΩ<br />
(6.20)<br />
and the density m<strong>at</strong>rix in this case is 1<br />
Ze−β(H−µN) . The oper<strong>at</strong>or H−µN is sometimes<br />
called the grand Hamiltonian. In this deriv<strong>at</strong>ion we have used the fact<br />
th<strong>at</strong> we can define energy eigenst<strong>at</strong>es for specific particle numbers, or [H, N] = 0.<br />
Th<strong>at</strong> condition is not necessary, however. We can use the partition function in<br />
the form (6.20) even if the Hamiltonian does not commute with the particle<br />
number oper<strong>at</strong>or, like in the theory of superconductivity. The density oper<strong>at</strong>or<br />
is used to calcul<strong>at</strong>e probabilities. The probability of finding a system with<br />
temper<strong>at</strong>ure T , chemical potential µ, and volume V in a st<strong>at</strong>e |x〉 is given by<br />
〈x| 1<br />
Ze−β(H−µN) |x〉.<br />
It is also possible to use other Legendre transform<strong>at</strong>ions. If V is the volume<br />
oper<strong>at</strong>or, one can show th<strong>at</strong>