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International Journal of Modern Physics B

WAVE FUNCTIONS WITH DISCRETE AND WITH CONTINUOUS SPECTRUM FOR QUANTUM DAMPED HARMONIC OSCILLATOR PERTURBED BY A SINGULARITY

Jeong-Ryeol Choi

Published Mar 20, 2004• 7 citations
Publisher

Abstract

The quantum states with discrete and continuous spectrum for the damped harmonic oscillator perturbed by a singularity have been investigated using invariant operator and unitary operator together. The eigenvalue of the invariant operator for ω0≤β/2 is continuous while for ω0>β/2 is discrete. The wave functions for ω0=β/2 expressed in terms of the Bessel function and for ω0<β/2 in terms of the Kummer confluent hypergeometric function. The convergence of the probability density is more rapid for over-damped harmonic oscillator than that of the other two cases due to the large damping constant.

DOI
10.1142/s0217979204024495

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The quantum states with discrete and continuous spectrum for the damped harmonic oscillator perturbed by a singularity have been investigated using invariant operator and unitary operator together. The eigenvalue of the invariant operator for ω0≤β/2 is continuous while for ω0>β/2 is discrete. The wave functions for ω0=β/2 expressed in terms of the Bessel function and for ω0<β/2 in terms of the Kummer confluent hypergeometric function. The convergence of the probability density is more rapid for over-damped harmonic oscillator than that of the other two cases due to the large damping constant.
Using the invariant operator method and the unitary transformation method together, we obtained discrete and continuous solutions of the quantum damped driven harmonic oscillator. The wave function of the underdamped harmonic oscillator is expressed in terms of the Hermite polynomial while that of the overdamped harmonic oscillator is expressed in terms of the parabolic cylinder function. The eigenvalues of the underdamped harmonic oscillator are discrete while that of the critically damped and the overdamped harmonic oscillators are continuous. We derived the exact phases of the wave function for the underdamped, critically damped and overdamped driven harmonic oscillator. They are described in terms of the particular solutions of the classical equation of motion.
We investigated the coherent states of nonconservative harmonic oscillator with a singular perturbation. The invariant operator represented in terms of lowering and raising operators. We confirmed that if the difference between two eigenvalues, α and β, of coherent states is much larger than unity, the states |α> and |β> are approximately orthogonal to each another. We calculated the expectation values of various quantities such as invariant operator, Hamiltonian and mechanical energy in coherent state. The mechanical energy of the system described by the Kanai–Caldirola Hamiltonian decreased exponentially depending on γ as time goes by in coherent state.
We derived the quantum solutions of a repulsive oscillator driven by a timedependent force and a repulsive oscillator perturbed by a singularity using unitary transformation method. The wave functions of the system perturbed by a singularity are expressed in terms of the Kummer confluent hypergeometric function while that of the driven system are expressed in terms of the Weber function. Since the repulsive oscillator is unbound, its spectrum of eigenvalues is continuous. We applied our general development to the repulsive Caldirola–Kanai oscillator.
In the preceding paper (arXiv: 0710.2724 [quant-ph]) we have constructed the general solution for the master equation of quantum damped harmonic oscillator, which is given by the complicated infinite series in the operator algebra level. In this paper we give the explicit and compact forms to solutions (density operators) for some initial values. In particular, the compact one for the initial value based on a coherent state is given, which has not been given as far as we know. Moreover, some related problems are presented.