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Journal of Physics A Mathematical and General

Harmonic oscillator with exponentially decaying mass

R. K. Colegrave, M. Sebawe Abdalla

Published Sep 1, 1981• 92 citations
Publisher

Abstract

The problem of a harmonic oscillator with varying mass parameter is reduced by canonical transformation to the corresponding constant mass problem and is solved in the case of an exponentially decaying mass. The constructed canonical Hamiltonian has time-independent eigenvalues and eigenvectors. The cases of undercritical and overcritical damping are considered in detail. The Green function is calculated and the behaviour of coherent states is discussed. The theory is related to the case of a cavity oscillator with a decaying field as in threshold laser operation. In particular, the energy of the field is considered.

DOI
10.1088/0305-4470/14/9/022

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The problem of a harmonic oscillator with varying mass parameter is reduced by canonical transformation to the corresponding constant mass problem and is solved in the case of an exponentially decaying mass. The constructed canonical Hamiltonian has time-independent eigenvalues and eigenvectors. The cases of undercritical and overcritical damping are considered in detail. The Green function is calculated and the behaviour of coherent states is discussed. The theory is related to the case of a cavity oscillator with a decaying field as in threshold laser operation. In particular, the energy of the field is considered.
In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.
A quantum-mechanical model of a damped harmonic oscillator (both with time-independent and time-dependent parameters) is studied in the framework of the linear Schr\"odinger equation with a Hermitian nonstationary Hamiltonian. Integrals of the motion of this equation and their eigenstates, including coherent states, are constructed. The influence of an external harmonic force to the time evolution of various average values calculated over coherent states is considered, including the resonance case. The specific symmetry of the Hamiltonian leading to the new concept of loss-energy states is discussed.
The authors study the evolution of SU(1,1) coherent states, which may represent squeezed vacuum states, in harmonic oscillators with time-dependent masses. Such oscillators may model a quantised electromagnetic field in a cavity where the field changes with time under the action of some reservoir. They consider three types of variable mass: a decaying mass, a pulsating mass, and a mass with damping and pulsation. They have found that in the most interesting cases involving pulsation, a high degree of squeezing (enhanced squeezing) may be obtained in the vicinity of the resonance where the pulsation frequency is twice the natural oscillator frequency.
The classical problem of a damped harmonic motion is formulated wave mechanically, with particular reference to the charge in a tuned circuit, and the electromagnetic field in a resonant cavity. Results in close agreement with the classical problem are obtained, without using perturbation theory. The classical frequency appears in the time dependence of the eigenfunctions and the classical damping appears as a decay in the eigenvalues. Zero point energy and thermal noise are discussed. The formulation is expected to break down, at very high frequencies, for tuned circuits and it is suggested that the same may be true for cavity oscillations and travelling electromagnetic waves.
The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.