TIME-DEPENDENT PERTURBATION METHODS
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TIME-DEPENDENT PERTURBATION METHODS book
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as
Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1)
where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′
may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.