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Springer eBooks

Linear Harmonic Oscillator: I Solution of the Schrödinger Equation and Relationship with the Classical Oscillator

Ajoy Kumar Ghatak, S. Lokanathan

Published Jan 1, 2004• 0 citations
Publisher
DOI
10.1007/978-1-4020-2130-5_7

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arXiv:1609.00248•Journal of the Korean Physical Society•Nov 1, 2016•8 citations
The harmonic oscillator with a time-dependent frequency has a family of linear quantum invariants for the time-dependent Schrödinger equation, which are determined by any two independent solutions to the classical equation of motion. Ermakov and Pinney have shown that a general solution to the time-dependent oscillator with an inverse cubic term can be expressed in terms of two independent solutions to the time-dependent oscillator. We explore the connection between linear quantum invariants and the Ermakov-Pinney solution for the time-dependent harmonic oscillator. We advance a novel method to construct Ermakov-Pinney solutions to a class of time-dependent oscillators and the wave functions for the time-dependent Schrödinger equation. We further show that the first and the second Pöschl-Teller potentials belong to a special class of exact time-dependent oscillators. A perturbation method is proposed for any slowly-varying time-dependent frequency.