Solution of the Cauchy Problem for a Time-Dependent Schoedinger Equation

Details Solution of the Cauchy Problem for a Time-Dependent Schoedinger Equation Solution of the Cauchy Problem for a Time-Dependent Schoedinger Equation

2007-11-05

arxiv.org/abs/0711.0559v4

Physics

Mathematical Physics

29 pages, 4 figures

Scientific paper

10.1063/1.2938698

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schroedinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner-Pollaczek polynomials, a Hankel-type integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a by-product.

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