Page 42 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

28 2 Quantum mechanical density matrix

theory. Consideration of time averaging will introduce a second level of averaging
into the theory. We are working here in the Schr¨odinger representation of operators.
Any time dependence which they have is explicit. In the study of equilibrium sys-
tems we will assume that the operators of interest are explicitly time independent
which is to say that they are time independent in the Schr¨odinger representation
(but not of course in the Heisenberg representation).
We suppose as in the classical case that in studying the macroscopic systems
usually of interest in statistical physics, we are interested in the time averages of
experimental observables, which we denote as
1 t+τ/2
¯ ¯ ¯
φ = φ(t )dt (2.2)
τ t−τ/2
The double bar emphasizes that, unlike the classical case, two kinds of averaging
are taking place. As in the classical case, we define equilibrium to be a situation in
which these averages are independent of τ for any τ> τ 0 . By the same arguments as
in the classical case, the averages are then independent of t as well. It is convenient,
as in the classical case, to move the origin of time to t =−τ/2 before taking the
limit τ →∞ so that we are interested in averages of the form
1 τ
¯
¯ ¯
φ = lim φ(t )dt (2.3)

τ→∞ τ
0
In the classical case, the analogous average was related to an integral over the
classical variables q, p. In the quantum case the corresponding average is over a
set of basis states of the Hilbert space defined briefly above. In particular, let ψ ν (q)
be a set of eigenstates of some complete set of commuting operators and expand
the wavefunction in terms of it

(q, t) = a ν (t)ψ ν (q) (2.4)
ν
It is possible in general to choose the ψ ν (q) to be orthonormal and we will do so.
Then the Schr¨odinger equation, expressed in terms of the coefficients a ν , becomes

3N
∗
i¯h(∂a ν /∂t) = H νν a ν where H νν = ψ Hψ ν d q. Inserting (2.4) into (2.3)
ν ν
and assuming that φ op is explicitly time independent gives
1 τ 1 τ
¯ ¯
¯

φ = lim dt φ(t ) = lim a (t)a ν (t)dt φ νν (2.5)
∗

ν
τ→∞ τ 0 ν,ν τ→∞ τ 0
in which

3N
∗
φ νν = ψ φ op ψ ν d q (2.6)

ν