Page 44 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

30 2 Quantum mechanical density matrix

where
( j)∗ ( j)
ρ ν ν = P j a (2.11)

ν a
ν
all j
These equations are equivalent to (2.9) and (2.7). This point of view was emphasized
in the book by Tolman. 1
Before proceeding to the analysis which leads to the quantum mechanical ana-
logue to the canonical distribution function, we note some facts which follow from
the definition of ρ ν ν . First, ρ ν ν is Hermitian since

1 τ 1 τ
∗
ρ = lim a ν (t)a ν (t) dt = lim a (t)a ν (t)dt = ρ νν (2.12)
∗
∗

ν ν ν
τ→∞ τ 0 τ→∞ τ 0
This means that, for some purposes, ρ can be regarded as an operator which is an
observable. Secondly, consider Trρ:
1 τ 1
τ 2
∗
Trρ = ρ νν = lim a (t)a ν (t)dt = lim |a ν (t)| dt
ν
τ→∞ τ τ→∞ τ
ν ν 0 0 ν
(2.13)
But

3N 2 ∗ 3N ∗ 2
d q| (q, t)| = a (t)a ν (t) d qψ (q)ψ ν (q) = |a ν (t)| = 1
ν ν
ν,ν ν
(2.14)
so
1 τ
Trρ = lim dt = 1 (2.15)
τ→∞ τ 0
The requirement that Trρ = 1 (independent of basis) looks a lot like the requirement
that a probability distribution be normalized and, in fact, in any given representation
ν one can see from the definition that ρ νν is the probability of finding the system in
the state ψ ν when a measurement of the observables associated with the quantum
numbers ν is made. But in an arbitrary basis, ρ νν can have off diagonal elements

which are complex and have no trivial probability interpretation.
To understand the fundamental equilibrium forms of the density matrix, we
proceed much as in the classical case to prove a quantum version of the Liouville
theorem and then argue on the basis of expected additive properties of large systems
for a canonical form for the density matrix. The quantum mechanical version of the
Liouville theorem is quite simple to obtain. First, one must decide how to define
the time dependence of the density matrix. In the present case, we choose to define
the time derivative of ρ νν as

1 τ d
dρ νν
= lim (a (t)a ν (t)) dt (2.16)
∗
dt τ→∞ τ 0 dt ν