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The Cauchy problem for a forced harmonic oscillator

R.M. Lopez* and S.K. Suslov**

School of Mathematical and Statistical Sciences and Mathematical, Computational, and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287–1804, U.S.A. e–mail:*rlopez14@asu.edu ; *sks@asu.edu

Recibido el 14 de agosto de 2009
Aceptado el 20 de agosto de 2009

Abstract

We construct an explicit solution of the Cauchy initial value problem for the one–dimensional Schrödinger 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.

Keywords: The Cauchy initial value problem; the Schrödinger equation; forced harmonic oscillator; Landau levels; the hypergeometric functions; the Hermite polynomials; the Charlier polynomials; Green functions; Fourier transform and its generalizations; the Heisenberg–Weyl group N (3).

Resumen

En el presente trabajo construimos una solución explícita unidimensional a la ecuación de Schrödinger con condiciones iniciales de Cauchy y con un operador Hamiltoniano dependiente del tiempo para el oscilador armónico forzado. La correspondiente función de Green (propagador) se deriva con aplicaciones de la transformada de Fourier generalizada y con una relación a las representaciones del grupo TV (3) de Heisenberg–Weyl, para un caso especial primero y después se extiende al caso general. Estudiamos por medio de un producto una extención de tres parámetros a la integral clásica de Fourier. Consideramos, como una aplicación, el movimiento de una partícula giratoria en un campo eléctrico y en un campo magnético perpendicularmente uniforme; evaluamos en términos de polinomios de Charlier una transición de amplitud entre los niveles de Landau. Además resolvemos una ecuación similar a la de difusión con valores iniciales.

Descriptores: Problema de valor inicial de Cauchy; ecuación de Schrödinger; osilador armónico forzado; niveles de Landau; funciones hipergeometricas; polinomios de Hermite; polinomios de Charlier; funciones de Green; transformada de Fourier y sus generalizaciones; el grupo Heisenberg–Weyl.

PACS: 45.20.–d; 02.30.–f; 02.30.Nw

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Acknowledgments

This paper is written as a part of the summer 2007 program on analysis of the Mathematical and Theoretical Biology Institute (MTBI) at Arizona State University. The MTBI/SUMS undergraduate research program is supported by The National Science Foundation (DMS–0502349), The National Security Agency (DOD–H982300710096), The Sloan Foundation, and Arizona State University. The authors are grateful (211) to Professor Carlos Castillo–Chávez for support and Ref. 8. We thank Professors George Andrews, George Gasper, Slim Ibrahim, Hunk Kuiper, Mizan Rahman, Svetlana Roudenko, and Hal Smith for valuable comments.

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