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Consider a quantum system described by a timedependent Hamiltonian
of the form
In the language of perturbation theory, is known as the unperturbed
Hamiltonian and describes a system of interest such as a molecule or
a condensedphase sample such as a pure liquid or solid or a solution.
is known as the perturbation, and it often describes an external
system, such as a laser field, that will be used to probe the energy levels
and other properties of .
We now seek a solution to the timedependent Schrödinger equation

(1) 
subject to an initial state vector
.
In order to solve the equation, we introduce a new state vector
related to
by

(2) 
The new state vector
is an equally valid representation of the
state of the system. In Chapter 10, we introduced the concept of pictures
in quantum mechanics and discussed the difference between the Schrödinger and
Heisenberg pictures. Eqn. (2) represents
yet another picture of quantum mechanics, namely the interaction picture.
Like the Schrödinger and Heisenberg pictures, the interaction picture is a
perfectly valid way of representing a quantum mechanical system. The interaction
picture can be considered as ``intermediate'' between the Schröginer picture,
where the state evolves in time and the operators are static, and the
Heisenberg picture, where the state vector is static and the operators evolve.
However, as we will see shortly,
in the interaction picture, both the state vector and the operators
evolve in time, however, the timeevolution is determined by the
perturbation . Eqn. (2)
specifies how to transform between the Schrödinger and
interaction picture state vectors. The transformation of operators
proceeds in an analogous fashion. If denotes an operator in the
Schrödinger picture, its representation in the interaction picture is
given by

(3) 
which is equivalent to an equation of motion of the form

(4) 
Substitution of Eqn. (2) into the
timedependent Schrödinger equation yields
According to Eqn. (3), the
is the interactionpicture
representation of the perturbation Hamiltonian, and we will denote this operator
as . Thus, the timeevolution of the state vector in the
interaction picture is given a Schrödinger equation of the form

(6) 
The initial condition to Eqn. (6),
is, according to Eqn. (2),
also
.
In the next section, we will develop an iterative solution to
Eqn. (6), which will reveal a rich structure
of the propagator for timedependent systems.
Next: Iterative solution for the
Up: Timedependent perturbation theory
Previous: Timedependent perturbation theory
Mark E. Tuckerman
20080427