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Bulletin of the Korean Chemical Society

Exact Wave Function of a Time-Dependent Harmonic Oscillator

Tae Jun Park

Published Feb 20, 2002• 3 citations
Publisher

Abstract

Increasing theoretical attention has been paid to solving exactly time-dependent quantum mechanical problems for their own mathematical interest and for applications.Among
DOI
10.5012/bkcs.2002.23.2.355

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By applying the standard analytical techniques of solving partial differential equations, we have obtained the exact solution in terms of the Fourier sine series to the time-dependent Schrödinger equation describing a quantum one-dimensional harmonic oscillator of time-dependent frequency confined in an infinite square well with the two walls moving along some parametric trajectories. Based upon the orthonormal basis of quasi-stationary wave functions, the exact propagator of the system has also been analytically derived. Special cases like (i) a confined free particle, (ii) a confined time-independent harmonic oscillator, and (iii) an aging oscillator are examined, and the corresponding time-dependent wave functions are explicitly determined. Besides, the approach has been extended to solve the case of a confined generalized time-dependent harmonic oscillator for some parametric moving boundaries as well.
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.