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Acta Physica Sinica

A time-dependent damped harmonic oscillator with a force quadratic in velocity

Bowen Huang, 首都师范大学物理系，北京 100037

Published Jan 1, 2003• 0 citations
Publisher

Abstract

A time-dependent damped harmonic oscillator with a nonconservative force quadratic in velocity (THOFQV) is described by an appropriate Lagrangian,and hence the general solution of THOFQV can be found.The solution of the Schrdinger equation for the system,such as the eigenfunctions and eigenvalues,is derived by utilizing these invariant operators.

DOI
10.7498/aps.52.271

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A time-dependent damped harmonic oscillator with a nonconservative force quadratic in velocity (THOFQV) is described by an appropriate Lagrangian,and hence the general solution of THOFQV can be found.The solution of the Schrdinger equation for the system,such as the eigenfunctions and eigenvalues,is derived by utilizing these invariant operators.
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.
Some recent works on the path-integral formulation of nonconservative forces quadratic in velocity are examined critically. It is argued that the ambiguity resulting from ordering the operators in the Lagrangian and the Hamiltonian for this force is more serious than it is usually believed, and that one can construct a number of Hermitian Hamiltonians all satisfying Ehrenfest's theorem, but forming a set of noncommuting operators. A similar problem is encountered in the ordering of the Lagrangian for the path-integral approach. This is in addition to a number of well-known difficulties such as violation of the position-velocity uncertainty, and the differences between the invariances of the equation of motion, and the Lagrangian. An alternative way is suggested in which the damping force is replaced by a purely time-dependent term which produces the same classical motion, thus avoiding the above-mentioned difficulties. Using this formulation, wave functions and time-dependent energies are found for a quadratically damped harmonic oscillator, and for a particle moving in a dissipative medium when the friction is proportional to an arbitrary power of velocity.
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.