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Journal of the Physical Society of Japan

Exactly Soluble Harmonic Oscillator for a Particular Form of Time and Coordinates-Dependent Mass

A. D. Jannussis, G. Karayannis, Παναγιώτης Παναγόπουλος, V. Papatheou, M. Symeonidis, D. Vavougios, P.D. Siafarikas, V. Zisis

Published Mar 1, 1984• 9 citations
Publisher

Abstract

An exact solution of Schrodinger equation is presented for the problem of a harmonic oscillator of mass varying with time and coordinates according to m(q,t) = m0/e-vt-k02q2. This case, compared with known types of equivalent interactions, leads again to a new one that relates the damped harmonic oscillator with velocity-depentent interaction problem.

DOI
10.1143/jpsj.53.957

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An exact solution of Schrodinger equation is presented for the problem of a harmonic oscillator of mass varying with time and coordinates according to m(q,t) = m0/e-vt-k02q2. This case, compared with known types of equivalent interactions, leads again to a new one that relates the damped harmonic oscillator with velocity-depentent interaction problem.
The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.