16th Symmetries in Science | |
Generalizations of the Ermakov system through the Quantum Arnold Transformation | |
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 |
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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.
【 预 览 】
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