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Journal of Shenyang Normal University

Invariants for time-dependent harmonic oscillators from real representation of solution

Wu Chuang

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The invariants for the time-dependent one-dimensional harmonic oscillator and the time-dependent two-dimensional harmonic oscillator in a static magnetic field are derived from the real representation of the exact solution to the equation of motion.Mathematically,the orthogonal functions invariant is the angular momentum of an isotropic time-dependent two-dimensional harmonic oscillator.Based on the invariants obtained here,the wave function for time-dependent two-dimensional harmonic oscillator in a static magnetic field in cylindrical coordinate is simply derived and the dynamical and geometrical phases are easily got by expressing the wave function as the superpositions of the wave functions of time-dependent two-dimensional harmonic oscillator in rectangular coordinate.

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Comment on “Wave functions of a time-dependent harmonic oscillator in a static magnetic field”

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Physica Scripta•May 18, 2006•7 citations
The invariants for the time-dependent one-dimensional harmonic oscillator and the time-dependent two-dimensional harmonic oscillator in a static magnetic field are derived from the real representation of the exact solution to the equation of motion.Mathematically,the orthogonal functions invariant is the angular momentum of an isotropic time-dependent two-dimensional harmonic oscillator.Based on the invariants obtained here,the wave function for time-dependent two-dimensional harmonic oscillator in a static magnetic field in cylindrical coordinate is simply derived and the dynamical and geometrical phases are easily got by expressing the wave function as the superpositions of the wave functions of time-dependent two-dimensional harmonic oscillator in rectangular coordinate.
We show that the procedure used by Ferreira et al. [Phys. Rev. A 66, 024103 (2002)] is not correct for the following reasons: (i) the invariant I(t)I(t)I(t) they derived does not satisfy the Liouville--Von Neuman equation. (ii) They found that the eigenvalues of I(t)I(t)I(t) are time dependent which should not be the case according to the Lewis-Riesenfeld theory. We give a correct procedure to find the solution of the system they considered, i.e., the Schr\"odinger equation for a two-dimensional harmonic oscillator with time-dependent mass and frequency in the presence of a static magnetic field.