The time-dependent quantum harmonic oscillator: a pedagogical approach via the Lewis-Riesenfeld dynamical invariant method

TL;DR

This paper pedagogically explores the time-dependent quantum harmonic oscillator using the Lewis-Riesenfeld invariant method, illustrating its applications in quantum state manipulation and dynamics in traps, with educational problem-solving examples.

Contribution

It provides a clear, educational presentation of solving the time-dependent quantum harmonic oscillator using the Lewis-Riesenfeld method, including practical examples and applications.

Findings

Wave functions for time-dependent oscillators derived

Transition probabilities for frequency jumps calculated

Dynamics in Paul traps analyzed

Abstract

In quantum mechanics courses, students often solve the Schr\"odinger 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.