### 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 language | English (US) |
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Pages (from-to) | 196-215 |

Number of pages | 20 |

Journal | Revista Mexicana de Fisica E |

Volume | 55 |

Issue number | 2 |

State | Published - 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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## Cite this

*Revista Mexicana de Fisica E*,

*55*(2), 196-215.