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Physica Scripta

Exact Quantum State for the Time-Dependent Repulsive Oscillator with a Driving Force and with a Singular Perturbation

Jeong-Ryeol Choi

Published Jan 1, 2004• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/70/5/001

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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.
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.
We derive quantum solutions of a generalized inverted or repulsive harmonic oscillator with arbitrary time-dependent mass and frequency using the quantum invariant method and linear invariants, and write its wave functions in terms of solutions of a second-order ordinary differential equation that describes the amplitude of the damped classical inverted oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum, the associated uncertainty relation, and the quantum correlations between coordinate and momentum. As a particular case, we apply our general development to the generalized inverted Caldirola–Kanai oscillator.
For a harmonic oscillator with time-dependent (positive) mass and frequency, a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time-dependent frequency, as well as operators. For a driven harmonic oscillator, a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given, as a generalization of previous results, in terms of the solution of classical equation of motion of the driven system. Thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system, provided the quantum states of the corresponding system of unit mass are given.
An exactly solvable model of a nonrelativistic quantum singular oscillator with a controlled potential is constructed. A position-dependent mass is introduced instead of a constant mass of the quantum system under consideration in such a way that it is possible to purposefully control the shape of the potential, turning it into a quantum well with different depths. It is shown that analytical expressions of the wave functions of such a quantum system are described through 2F1 type hypergeometric functions and Jacobi polynomials, whereas, the discrete energy spectrum corresponding to these analytic solutions is non-equidistant. In a specific limit, when the above-mentioned control of the potential shape is lost, the wave functions accurately restore the wave functions of an ordinary quantum singular oscillator, which are described through generalized Laguerre polynomials. At this limit, the energy spectrum also restores the usual equidistant energy spectrum of a quantum singular oscillator.