### 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) |
---|---|

Pages (from-to) | 196-215 |

Number of pages | 20 |

Journal | Revista Mexicana de Fisica E |

Volume | 55 |

Issue number | 2 |

State | Published - Dec 2009 |

### Fingerprint

### 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 hypergeometric functions
- The Schrödinger equation

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Education

### Cite this

*Revista Mexicana de Fisica E*,

*55*(2), 196-215.

**The Cauchy problem for a forced harmonic oscillator.** / Lopez, R. M.; Suslov, Sergei.

Research output: Contribution to journal › Article

*Revista Mexicana de Fisica E*, vol. 55, no. 2, pp. 196-215.

}

TY - JOUR

T1 - The Cauchy problem for a forced harmonic oscillator

AU - Lopez, R. M.

AU - Suslov, Sergei

PY - 2009/12

Y1 - 2009/12

N2 - 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.

AB - 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.

KW - Forced harmonic oscillator

KW - Fourier transform and its generalizations

KW - Green functions

KW - Landau levels

KW - The Cauchy initial value problem

KW - The Charlier polynomials

KW - The Heisenberg-Weyl group N(3)

KW - The Hermite polynomials

KW - The hypergeometric functions

KW - The Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=80053108043&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80053108043&partnerID=8YFLogxK

M3 - Article

VL - 55

SP - 196

EP - 215

JO - Revista Mexicana de Fisica

JF - Revista Mexicana de Fisica

SN - 0035-001X

IS - 2

ER -