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Journal of Physics A Mathematical and Theoretical

Eigenstates of the higher powers of an annihilation operator for a time-dependent harmonic oscillator and their properties

Xiao-Hua Wang, Jin-Song Huang, Yu Liu, Lian-Fu Wei

Published Mar 3, 2010• 0 citations
Publisher

Abstract

In the framework of dynamical invariant theory, we introduce the annihilation operator and creation operator for a time-dependent harmonic oscillator. The orthonormalized eigenstates of the operator are constructed, and their mathematical and quantum statistical properties are discussed in detail. Specifically, a particle moving in a Paul trap with periodically varying frequency is treated as an example to demonstrate the dynamical squeezing effects for the eigenstates of the operator .

DOI
10.1088/1751-8113/43/12/125302

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In the framework of dynamical invariant theory, we introduce the annihilation operator and creation operator for a time-dependent harmonic oscillator. The orthonormalized eigenstates of the operator are constructed, and their mathematical and quantum statistical properties are discussed in detail. Specifically, a particle moving in a Paul trap with periodically varying frequency is treated as an example to demonstrate the dynamical squeezing effects for the eigenstates of the operator .
The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time-dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.