Time evolution 399=
∑
lwl|〈ak|wl〉|^2. (10.2.13)Equating the results of eqns. (10.2.12) and (10.2.13) gives
1
Z∑Z
λ=1〈Pa(λk)〉=∑
lwl|〈ak|wl〉|^2. (10.2.14)We now interpret{|wl〉}as a complete set of microscopic states appropriate for the
ensemble, withwlthe probability that a randomly selected member of the ensemble
is in the state|wl〉. Hence, the quantity on the right is the sum of probabilities that
a measurement ofAˆin a state|wl〉yields the resultakweighted by the probability
that an ensemble member is in the state|wl〉. This is equal to the ensemble averaged
probability on the left. Thus, the density operator ̃ρgives the probabilitywlfor an
ensemble member to be in a particular microscopic state|wl〉consistent with a set of
macroscopic observables, and therefore, it plays the same role in quantum statistical
mechanics as the phase space distribution functionf(x) plays in classical statistical
mechanics.
10.3 Time evolution of the density matrix
The evolution in time of the density matrix is determined by the time evolution of
each of the state vectors|Ψ(λ)〉. The latter are determined by the time-dependent
Schr ̈odinger equation. Starting from eqn. (10.2.6), we write the time-dependent density
operator as
ρˆ(t) =∑Z
λ=1|Ψ(λ)(t)〉〈Ψ(λ)(t)|. (10.3.1)An equation of motion for ˆρ(t) can be determined by taking the time derivative of
both sides of eqn. (10.3.1):
∂ρˆ
∂t=
∑Z
λ=1[(
∂
∂t|Ψ(λ)(t)〉)
〈Ψ(λ)(t)|+|Ψ(λ)(t)〉(
∂
∂t〈Ψ(λ)(t)|)]
. (10.3.2)
However, since∂|Ψ(λ)(t)〉/∂t= (1/i ̄h)Hˆ|Ψ(λ)(t)〉from the Schr ̈odinger equation, eqn.
(10.3.2) becomes
∂ρˆ
∂t=
1
i ̄h∑Z
λ=1[(
Hˆ|Ψ(λ)(t)〉)
〈Ψ(λ)(t)|−|Ψ(λ)(t)〉(
〈Ψ(λ)(t)|Hˆ)]
=
1
i ̄h(Hˆρˆ−ρˆHˆ) (10.3.3)or
∂ρˆ
∂t
=
1
i ̄h[Hˆ,ρˆ]. (10.3.4)Eqn. (10.3.4) is known as thequantum Liouville equation, and it forms the basis
of quantum statistical mechanics just as the classical Liouville equation derived in
Section 2.5 forms the basis of classical statistical mechanics.