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Time evolution of the density operator

The time evolution of the operator tex2html_wrap_inline509 can be predicted directly from the Schrödinger equation. Since tex2html_wrap_inline599 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 tex2html_wrap_inline601 is


Note that the equation of motion for tex2html_wrap_inline599 differs from the usual Heisenberg equation by a minus sign! Since tex2html_wrap_inline599 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 tex2html_wrap_inline599 can be cast into a quantum Liouville equation by introducing an operator


In term of iL, it can be seen that tex2html_wrap_inline599 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 tex2html_wrap_inline509 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 tex2html_wrap_inline617 , which satisfies


with tex2html_wrap_inline619 being the classical Liouville operator. Again, we see that the limit of a commutator is the classical Poisson bracket.

Mark Tuckerman
Tue May 9 19:40:24 EDT 2000