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Journal of Russian Laser Research

Exact wave functions for generalized harmonic oscillators

Nathan A. Lanfear, Raquel M. López, Sergeĭ K. Suslov

Published Jul 1, 2011• 39 citations
Publisher
DOI
10.1007/s10946-011-9223-1

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The unitary operator that transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different timescale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.
Using the trial function method, exact wave functions of the time-dependent driven generalized harmonic oscillator (TDGHO) were derived. The wave functions are wave packets with the center moving according to the classical equation of motion. Correspondingly, the geometrical phase is divided into two parts with one part generated by the shape change of the wave packet and the other part by the motion of the wave packet center. The exact classical solution as well as the classical phase was obtained from a quantum description by the Heisenberg correspondence principle. Finally, the relationship between the quantum and classical phases was achieved.