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American Journal of Physics

Perturbative treatment of a general driven time-dependent oscillator in a time-dependent basis

Chi‐Fai Lo

Published Mar 1, 1991• 5 citations
Publisher

Abstract

This paper examines the possibility of applying the time-dependent perturbation theory to the systems with a time-dependent unperturbed part. Analyses indicate that time-dependent perturbation theory is valid even if the unperturbed Hamiltonian H0 depends on time, provided that the pseudostationary states of H0 are known. It is also shown that this new perturbative treatment provides a useful tool for tackling the problem of a general driven time-dependent oscillator.

DOI
10.1119/1.16572

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This paper examines the possibility of applying the time-dependent perturbation theory to the systems with a time-dependent unperturbed part. Analyses indicate that time-dependent perturbation theory is valid even if the unperturbed Hamiltonian H0 depends on time, provided that the pseudostationary states of H0 are known. It is also shown that this new perturbative treatment provides a useful tool for tackling the problem of a general driven time-dependent oscillator.
We explore how to apply perturbation theory to complicated time-dependent Hamiltonian systems that involve complex potentials. To do this, we introduce a generalized time-dependent oscillator to which the complex potentials are connected through a weak coupling strength. We regard the complex potentials in the Hamiltonian as the perturbed terms. Quantum characteristics of the system, such as wave functions and expectation values of the Hamiltonian, are investigated on the basis of the perturbation theory. We apply our theory to particular systems with explicit choices of time-dependent parameters. Through such applications, the time behavior of the quantum wave packets and the spectrum of expectation values of the Hamiltonian are analyzed in detail. We confirm that the imaginary parts of expectation values of the Hamiltonian are not zero but very small, while the real parts deviate slightly from those of the unperturbed system.
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.