Generalizations of the Ermakov system through the Quantum Arnold Transformation

16th Symmetries in Science

López-Ruiz, Francisco F.^1 ; Guerrero, Julio^2

Departamento de Física Aplicada, Universidad de Cádiz, Campus de Puerto Real, Puerto Real, Cádiz

11510, Spain^1

Departamento de Matemática Aplicada, Universidad de Murcia, Campus de Espinardo, Murcia

30100, Spain^2

关键词: Arnold transformation;    Constant of motion;    Ermakov-Pinney equation;    Harmonic oscillators;    Nonlinear differential equation;    Physical interpretation;    Time dependent frequency;    Unitary operators;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/538/1/012015/pdf
DOI  :  10.1088/1742-6596/538/1/012015

来源: IOP

【 摘 要 】

An Ermakov system consists of a pair of coupled non-linear differential equations which share a joint constant of motion named Ermakov invariant. One of those equations, non-linear, is frequently referred to as the Ermakov-Pinney equation; the other equation may be thought of as describing a dynamical system: a harmonic oscillator with time-dependent frequency. In this paper, we revise the Quantum Arnold Transformation, a unitary operator mapping the solutions of the Schrödinger equation for time-dependent (even damped) harmonic oscillators, described by the Generalized Caldirola-Kanai equation, into solutions for the free particle. With this tool, we elucidate the existence of Ermakov-type invariants in classically linear systems at the classical and quantum levels. We also provide more general Ermakov-type systems and the corresponding invariants, together with a physical interpretation.