Page 46 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

32 2 Quantum mechanical density matrix

For large systems, partitionable in the same sense discussed for classical systems,
we can now construct arguments for a canonical density matrix very similar to
those in the last chapter. In particular, we suppose that for a partition into two
large systems, the density matrix is a product in the following sense. We first
work formally in the special basis discussed in the last paragraph, in which the
density matrix is rigorously diagonal and in which the quantum numbers {ν} denote
eigenvectors of 3N linearly independent operators, including H which commute
with the Hamiltonian H. Because ρ itself commutes with the Hamiltonian it must
itself be diagonal in this representation. If the Hamiltonian of the partitioned system
can, to a good approximation (and using arguments completely analogous to those
used in the classical case), be written as H 1 + H 2 , ignoring interaction terms in
the thermodynamic limit, then in this representation the diagonal elements of the
density matrix (which are the only nonzero ones) can be written
ρ ν,ν = ρ (1) ρ (2) (2.22)
ν 1 ,ν 1 ν 2 ,ν 2
where ν 1 and ν 2 designate bases for the two partitions which also simultaneously
diagonalize all the constants of the motion of those two partitions individually. Now
we may take the natural logarithm of (2.22) much as in the classical case:

ln ρ ν,ν = ln ρ (1) + ln ρ (2) (2.23)
ν 1 ,ν 1 ν 2 ,ν 2
which shows that ln ρ ν,ν can be a function only of the quantum numbers in ν
corresponding to additive constants of the motion. If, as in the classical case, we
suppose these to be energy, linear and angular momentum then we have

ρ νν = δ ν,ν e e (2.24)
α −βE ν + δ· P ν + γ · L ν

in this representation. Here E ν , P ν , L ν are the eigenvalues of energy, linear and

angular momentum. We may determine α from the requirement that Trρ = 1:
e −βE ν + δ· P ν + γ · L ν
(2.25)

ρ νν = δ ν,ν
e −βE ν + δ· P ν + γ · L ν
ν
Finally one can remove the restriction to a particular basis, noting that (2.25) can
be written as the operator
e −βH+ δ· P+ γ · L
ρ =
(2.26)
Tr e −βH+ δ· P+ γ · L
and if we restrict attention to the case of systems in a stationary “box”

e −βH
ρ = (2.27)
Tr(e −βH )
which is called the canonical density matrix.