Statistical Mechanics - Physics at Oregon State University

126 CHAPTER 6. DENSITY MATRIX FORMALISM.
is defined. We restrict the quantum mechanical wave functions to states with
a definite volume V and number of particles N. The energy restriction tells us
that we only want to consider states with 〈x| H |x〉 = U. The Hilbert space of
all possible states |x〉 obeying these conditions is SU,V,N .
The problem of the microcanonical ensemble is therefore to maximize the
entropy (6.29) in this space. hence we want to find the maximal value of the
entropy for all operators ρ obeying the conditions (6.4) through (6.6). Limiting
our search to Hermitian operators only is easy. The condition (6.6) is an
inequality and very hard to incorporate in general during the search. Therefore,
this condition we will always have to test at the end. We find all possible
maxima of the entropy, and throw away those that correspond to an operator
not obeying (6.6). Finally, the equality condition (6.4) is incorporated via a
Lagrange multiplier.
The task is now to find the maximum of
X(ρ) = −kBTr ρ log(ρ) + λkB(Tr ρ − 1) (6.31)
over all Hermitian operators on the space SU,V,N . It turns out to be convenient
to extract a factor kB from the Lagrange multiplier. Based on this euqation,
we need some functional derivative equation of the form δX
δρ = 0, but now using
operators. How do we define that? Suppose we change the operator by a small
amount, keeping it Hermitian. The we define
∆X = X(ρ + ∆ρ) − X(ρ) (6.32)
just as we did for functional derivatives. In order to connect with what we did
before, assume that we have some basis in the space on which the operators are
acting. Define the density matrix by
and the variation by
In first order we can then write
Rij = 〈i| ρ |j〉 (6.33)
∆Rij = 〈i| ∆ρ |j〉 (6.34)
∆X =
and now we define the operator δX
δρ
This leads to
∆X =
i,j
〈j| δX
δρ
〈j| δX
δρ
i,j
Aji∆Rij
by its matrix elements:
|i〉 = Aji
(6.36)
(6.35)
|i〉 〈i| ∆ρ |j〉 = Tr (δX ∆ρ) (6.37)
δρ