Construction of Exact Ermakov-Pinney Solutions and Time-Dependent Quantum Oscillators
Abstract
The harmonic oscillator with a time-dependent frequency has a family of linear quantum invariants for the time-dependent Schr\"{o}dinger 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 Schr\"{o}dinger equation. We further show that the first and the second P\"{o}schl-Teller potentials belong to a special class of exact time-dependent oscillators. A perturbation method is proposed for any slowly-varying time-dependent frequency.
Keywords
Cite
@article{arxiv.1609.00248,
title = {Construction of Exact Ermakov-Pinney Solutions and Time-Dependent Quantum Oscillators},
author = {Sang Pyo Kim and Won Kim},
journal= {arXiv preprint arXiv:1609.00248},
year = {2016}
}
Comments
RevTex 7 pages, no figure; perturbation method, connection to short-cuts to adiabaticity and references are added