The time evolution of the operator can be predicted directly from the Schrödinger equation. Since is given by

the time derivative is given by

where the second line follows from the fact that the Schrödinger equation for the bra state vector is

Note that the equation of motion for differs from the
usual Heisenberg equation by a minus sign! Since is
constructed from state vectors, it is not an observable like
other hermitian operators, so there is no reason to expect that
its time evolution will be the same. The general solution to
*its* equation of motion is

The equation of motion for can be cast into a quantum Liouville equation by introducing an operator

In term of *iL*, it can be seen that satisfies

What kind of operator is *iL*? It acts on an operator and returns another
operator. Thus, it is not an operator in the ordinary sense, but is
known as a *superoperator* or *tetradic operator*
(see S. Mukamel, *Principles of Nonlinear Optical Spectroscopy*, Oxford University
Press, New York (1995)).

Defining the evolution equation for this way, we have a perfect analogy between the density matrix and the state vector. The two equations of motion are

We also have an analogy with the evolution of the classical phase space distribution , which satisfies

with being the classical Liouville operator. Again, we see that the limit of a commutator is the classical Poisson bracket.

Tue May 9 19:40:24 EDT 2000