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Journal of Physics A Mathematical and General

Damping of wave packet motion in a general time-dependent quadratic field

B. Rémaud, E. Susana Hernández

Published Jun 1, 1980• 41 citations
Publisher

Abstract

The authors provide a framework for the study of a quantal time-dependent oscillator in the presence of a loss mechanism. Previous approaches to partial aspects of the problem are analysed and cast into a unified global picture. Several alternative descriptions of the situation are analysed and it is shown that the proper generalisation of the Hamiltonian formulation by Kostin (1975) is adequate for the case under consideration. Applications to a number of problems in which the mass of the oscillator is a given function of time are presented, including cases in which the mass becomes infinite.

DOI
10.1088/0305-4470/13/6/024

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The authors provide a framework for the study of a quantal time-dependent oscillator in the presence of a loss mechanism. Previous approaches to partial aspects of the problem are analysed and cast into a unified global picture. Several alternative descriptions of the situation are analysed and it is shown that the proper generalisation of the Hamiltonian formulation by Kostin (1975) is adequate for the case under consideration. Applications to a number of problems in which the mass of the oscillator is a given function of time are presented, including cases in which the mass becomes infinite.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.
The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.
Using an algebraic approach, it is possible to obtain the temporal evolution wave function for a Gaussian wave-packet obeying the quadratic time-dependent Hamiltonian (QTDH). However, in general, most of the practical cases are not exactly solvable, for we need general solutions of the Riccatti equations which are not generally known. We therefore bypass directly solving for the temporal evolution wave function, and study its inverse problem. We start with a particular evolution of the wave-packet, and get the required Hamiltonian by using the inverse method. The inverse approach opens up a new way to find new exact solutions to the QTDH. Some typical examples are studied in detail. For a specific time-dependent periodic harmonic oscillator, the Berry phase is obtained exactly.