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TURKISH JOURNAL OF PHYSICS

A Generalized Oscillator System

M. Kibler, Levon Grigory Mardoyan, GEORGE S. POGOSYAN

Published Jan 1, 1997• 2 citations
Publisher

Abstract

This article deals with a nonrelativistic quantum mechanical study of a dynamical system which generalizes the isotropic harmonic oscillator system in three dimensions. The quantum mechanical spectum of this system is worked out in some details. The problem of interbasis expansions of the wavefunctions is completely solved
DOI
10.55730/1300-0101.2376

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In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.
We investigate the dynamics of the spin-less relativistic particle subject to an external eld of a harmonic oscillator potential.The KleinGordon equation with one-and three-dimensional vector and scalar parabolic potentials is solved using the expansion of the wavefunction in properly selected basis-sets.The resonance states are determined using the complex coordinate rotation method.The analytic expressions for the rst-and secondorder relativistic energy corrections are derived perturbatively.The relativistic model of the harmonic oscillator in the momentum representation, originally proposed by Znojil, is also discussed.
Complete description of the classical and quantum dynamics of a particle in an anisotropic, rotating, harmonic trap is given. The problem is studied in three dimensions and no restrictions on the geometry are imposed. In the generic case, for an arbitrary orientation of the rotation axis, there are two regions of instability with different characteristics. The analysis of the quantum-mechanical problem is made simple due to a direct connection between the classical mode vectors and the quantum-mechanical wave functions. This connection is obtained via the matrix Riccati equation that governs the time evolution of squeezed states of the harmonic oscillator. It is also shown that the inclusion of gravity leads to a resonant behavior -- the particles are expelled from the trap.
Chapter 9 is devoted to a detailed study of the harmonic oscillator in one dimension. The time-dependent Schrödinger equation is used to construct the coherent oscillator packet with minimal dispersion, which provides an opportunity to appreciate the similarities and differences between the dynamics of the classical and quantum oscillators. In a second step, we study the stationary solutions of the oscillator, which represent one of the most important orthogonal sets of quantum eigenfunctions because of its many applications, properties, and relative simplicity. Further, the use of raising and lowering (or creation and annihilation) operators is shown to be a powerful way of dealing with both material and field oscillators in general. Finally, the problem of two coupled oscillators is used to generalize the Schrödinger equation to the case of more than one particle.