The Cauchy problem for a forced harmonic oscillator

R. M. Lopez, Sergei Suslov

Research output: Contribution to journalArticle

11 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)196-215
Number of pages20
JournalRevista Mexicana de Fisica E
Volume55
Issue number2
StatePublished - Dec 2009

Keywords

  • Forced harmonic oscillator
  • Fourier transform and its generalizations
  • Green functions
  • Landau levels
  • The Cauchy initial value problem
  • The Charlier polynomials
  • The Heisenberg-Weyl group N(3)
  • The Hermite polynomials
  • The Schrödinger equation
  • The hypergeometric functions

ASJC Scopus subject areas

  • Education
  • Physics and Astronomy(all)

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