400 Quantum ensembles
Recall that the time evolution of a Hermitian operator representinga physical
observable in the Heisenberg picture is given by eqn. (9.2.56). Although ˆρis a Hermi-
tian operator, its evolution equation differs from eqn. (9.2.56), as eqn. (10.3.4) makes
clear. This difference underscores the fact that ˆρdoes not actually represent a physical
observable.
The quantum Liouville equation can be solved formally as
ˆρ(t) = e−i
Hˆt/ ̄h
ρˆ(0)ei
Hˆt/ ̄h
=U(t)ˆρ(0)U†(t). (10.3.5)Eqn. (10.3.4) is often cast into a form that closely resembles the classical Liouville
equation by defining a quantum Liouville operator
iL=1
i ̄h[...,Hˆ]. (10.3.6)
In terms of this operator, the quantum Liouville equation becomes
∂ρˆ
∂t=−iLρ,ˆ (10.3.7)which has the formal solution
ρˆ(t) = e−iLtρˆ(0). (10.3.8)
There is a subtlety associated with the quantum Liouville operatoriL. As eqn. (10.3.6)
implies,iLis not an operator in the sense described in Section 9.2. The operators we
have encountered so far act on the vectors of the Hilbert space to yield new vectors.
By contrast,iLacts on an operator and returns a new operator. For this reason, it is
often called a “superoperator” or “tetradic” operator.^1
10.4 Quantum equilibrium ensembles
As in the classical case, quantum equilibrium ensembles are defined bya density matrix
with no explicit time dependence, i.e.∂ρ/∂tˆ = 0. Thus, the equilibrium Liouville
equation becomes [Hˆ,ρˆ] = 0. This is precisely the condition required for a quantity to
be a constant of the motion. The general solution to the equilibrium Liouville equation
is any functionF(Hˆ) of the Hamiltonian. Consequently,Hˆand ˆρhave simultaneous
eigenvectors. If|Ek〉are the eigenvectors ofHˆ with eigenvaluesEk, then
ρˆ|Ek〉=F(Hˆ)|Ek〉=F(Ek)|Ek〉. (10.4.1)Starting from eqn. (10.4.1), we could derive the quantum equilibrium ensembles in
much the same manner as we did for the classical equilibrium ensembles. That is, we
could begin by defining the microcanonical ensemble based on the conservation ofHˆ
and then derive the canonical, isothermal-isobaric, and grand canonical ensembles by
coupling the system to a heat bath, mechanical piston, particle reservoir, etc. How-
ever, since we have already carried out this program for the classical ensembles, we
(^1) As an example from the literature of the use of the superoperator formalism, S. Mukamel, in
his bookPrinciples of Nonlinear Optical Spectroscopy(1995), uses the quantum Liouville operator
approach to develop an elegant framework for analyzing various types of nonlinear spectroscopies.