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8.5: Dyson Series

Let us now try to find approximate solutions of Equation (749) for a general system. It is convenient to work in terms of the time evolution operator, , which is defined

 (777)

Here, is the state ket of the system at time , given that the state ket at the initial time is . It is easily seen that the time evolution operator satisfies the differential equation

 (778)

subject to the initial condition

 (779)

In the absence of the external perturbation, the time evolution operator reduces to

 (780)

Let us switch on the perturbation and look for a solution of the form

 (781)

It is readily demonstrated that satisfies the differential equation

 (782)

where

 (783)

subject to the initial condition

 (784)

Note that specifies that component of the time evolution operator which is due to the time-dependent perturbation. Thus, we would expect to contain all of the information regarding transitions between different eigenstates of caused by the perturbation.

Suppose that the system starts off at time in the eigenstate of the unperturbed Hamiltonian. The subsequent evolution of the state ket is given by Equation (744),

 (785)

However, we also have

 (786)

It follows that

 (787)

where use has been made of . Thus, the probability that the system is found in state at time , given that it is definitely in state at time , is simply

 (788)

This quantity is usually termed the transition probability between states and .

Note that the differential equation (782), plus the initial condition (784), are equivalent to the following integral equation,

 (789)

We can obtain an approximate solution to this equation by iteration:

This expansion is known as the Dyson series. Let

 (791)

where the superscript refers to a first-order term in the expansion, etc. It follows from Equations (787) and (790) that

 (792) (793) (794)

These expressions simplify to

 (795) (796) (797)

where

 (798)

and

 (799)

The transition probability between states and is simply

 (800)

According to the above analysis, there is no chance of a transition between states and (where ) to zeroth order (i.e., in the absence of the perturbation). To first order, the transition probability is proportional to the time integral of the matrix element , weighted by some oscillatory phase-factor. Thus, if the matrix element is zero then there is no chance of a first-order transition between states and . However, to second order, a transition between states and is possible even when the matrix element is zero.