Abstract

LOPEZ, R.M.  and  SUSLOV, S.K.. The Cauchy problem for a forced harmonic oscillator. Rev. mex. fís. E [online]. 2009, vol.55, n.2, pp.196-215. ISSN 1870-3542.

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).

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