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Journal of Physics A Mathematical and General

Quantum treatment of a class of time-dependent potentials

I. H. Duru

Published Nov 21, 1989• 23 citations
Publisher

Abstract

The time-dependent potential V(x-f(t)) is studied by path integrals. It is shown that the problem can be mapped into the static form of the potential plus a linear term with a time-dependent coefficient. After the presentation of the general formulation, some exactly solvable examples are discussed. A perturbative treatment is also suggested.

DOI
10.1088/0305-4470/22/22/013

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The time-dependent potential V(x-f(t)) is studied by path integrals. It is shown that the problem can be mapped into the static form of the potential plus a linear term with a time-dependent coefficient. After the presentation of the general formulation, some exactly solvable examples are discussed. A perturbative treatment is also suggested.
A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.
A class of time-dependent problems in two space dimensions possessing time-dependent invariants, bilinear in momenta, is considered. Explicit expression of the potentials and the corresponding invariants are derived. Quantum mechanics is introduced in these time-dependent problems directly through a Feynman propagator defined as a path integral involving the classical action. The propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators. Equivalence of the present theory to that of Lewis and Riesenfeld [H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)] is discussed.
Within the framework of Feynman path integration, expectation values of quantum mechanical operators may be exactly obtained for a class of time-dependent problems. Attention is focused on the two-dimensional motion of a charged particle in a perpendicular magnetic field with a time-dependent driving force. A harmonic oscillator potential is included to ensure that the corresponding density matrix is properly defined although some expectation values are defined without it. This potential is at least a mathematical convenience. Some discussion concerning the conditions under which steady states may be attained is also included.