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Journal of Mathematical Physics

Exact solution of a time-dependent quantal harmonic oscillator with damping and a perturbative force

Dinkar C. Khandekar, S. V. Lawande

Published Sep 1, 1979• 85 citations
Publisher

Abstract

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.

DOI
10.1063/1.524305

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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.
Summary. - We obtain here the solution of a system of two coupled oscillators under the influence of damping force and time-dependent external force. The solution is obtained by first constructing linear invariants for this system. The exact dynamics of time-dependent quantal harmonic oscillator with timedependent frequency has been extensively studied in recent years exploiting the explicitly time-dependent invariants (1-5). The fact that there exists a simple relationship between the eigenstates of the invariant and the solution of the corresponding SehrSclinger equation has prompted people to find general form of invariants for time-dependent oscillator-type Hamiltonian. In these approaches quadratic type of invariants have mostly been considered. Further, these invariants involve a simple function ~(t), related to the amplitude of the classical free oscillator, and satisfies a nonlinear equation which should be solved for practical application. However, there has been attempt so far to study the dynamics of a time-dependent coupled-oscillator system. We, therefore, consider in the present work a system of two coupled oscillators, both damped and under time-dependent external force. To get the solution of the SchrSdinger equation for this system, we first obtain explicitly timedependent invariants which are linear in position and momentum variables. The two advantages with the linear invariants is that the eigenvalues of the invariant can be made continuous with suitable initial condition on the motion of oscillators and, further, that it involves only the amplitude of the classical oscillator and thus can be easily diagonalized.
Explicitly time-dependent invariants, linear in x and p, are found to simplify the solution of the Schrödinger equation with oscillator-type Hamiltonian. As an example, exact dynamics is obtained of the time-dependent quantal oscillator, with damping and subject to external force. The invariant and hence the solution of the Schrödinger equation involve only the amplitude of the classical damped oscillator. The identity of our solution with a recently obtained result is illustrated. A digressive remark on quadratic invariants and the Dirac operator formalism is added.
The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.
For a damped harmonic oscillator forced by a time-dependent field, the exact wave function is obtained by three different methods: (i) path integral, (ii) second quantization, and (iii) dynamical invariant. The explicit form of the dynamical invariant involves a solution to a corresponding auxiliary equation. The coherent states, defined as eigenstates of a new destruction operator, form a nonorthogonal, overcomplete set and correspond to the minimum uncertainty states. These coherent states give the exact classical motion of the damped driven harmonic oscillator.
The quantal problem of a particle interacting in one dimension with an external time-dependent quadratic potential and a constant inverse square potential is exactly solved. The solutions are found both in the Schrödinger representation, by using a generating function or a time-dependent raising operator, and in the Heisenberg picture. They depend only on the solution of the classical harmonic oscillator. The generalizations to the n-dimensional problem and to the problem of N particles in one dimension, interacting pairwise via a quadratic time-dependent potential and a constant inverse square potential, are finally sketched.