The Cauchy Problem for a Forced Harmonic Oscillator

TL;DR

This paper derives explicit solutions for the time-dependent Schrödinger equation of a forced harmonic oscillator, using Fourier transforms and group representations, and explores applications to quantum particle motion and diffusion equations.

Contribution

It provides a novel explicit solution method for the forced harmonic oscillator's quantum dynamics and extends Fourier analysis with a three-parameter integral, linking to physical applications.

Findings

Explicit Green function for the Schrödinger equation derived

Transition amplitudes between Landau levels evaluated

Extension of Fourier integral with three parameters discussed

Abstract

We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schroedinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg-Weyl group N(3) in a certain special case first, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric fields is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.