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arXiv · 0707.1902

The Cauchy Problem for a Forced Harmonic Oscillator

Raquel M. Lopez·Sergei K. Suslov
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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.

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Raquel M. Lopez, Sergei K. Suslov. 2007-12-27. The Cauchy Problem for a Forced Harmonic Oscillator. https://arxiv.org/abs/0707.1902

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