Time-Dependent Diffeomorphisms as Quantum Canonical Transformations and the Time-Dependent Harmonic Oscillator
摘要
Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schr\"odinger equation. A simple generalization of this result leads to the reduction of the Schr\"odinger equation to a second order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. Time-evolution operator is expressed in terms of a positive real solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.
引用
@article{arxiv.quant-ph/9807002,
title = {Time-Dependent Diffeomorphisms as Quantum Canonical Transformations and the Time-Dependent Harmonic Oscillator},
author = {Ali Mostafazadeh},
journal= {arXiv preprint arXiv:quant-ph/9807002},
year = {2008}
}
备注
Plain Latex, J. Phys. A: Math. Gen., to appear