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Journal of Physics A Mathematical and Theoretical

Generalized Lippmann–Schwinger equation in the fractional quantum mechanics

Jianping Dong

Published Apr 26, 2011• 8 citations
Publisher

Abstract

From the three-dimensional space fractional Schrödinger equation, a generalized Lippmann-Schwinger equation for the fractional quantum mechanics is obtained for both scattering and bound states.We apply the generalized integral equation to study the fractional quantum scattering problem and give the approximate scattering wavefunction of first order and higher orders.

DOI
10.1088/1751-8113/44/21/215204

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arXiv:1910.03682•Oct 8, 2019•0 citations
From the three-dimensional space fractional Schrödinger equation, a generalized Lippmann-Schwinger equation for the fractional quantum mechanics is obtained for both scattering and bound states.We apply the generalized integral equation to study the fractional quantum scattering problem and give the approximate scattering wavefunction of first order and higher orders.
The fractional Schrödinger equation is solved for a free particle and for an infinite square potential well. The fundamental solution of the Cauchy problem for a free particle, the energy levels and the normalized wave functions of a particle in a potential well are obtained. In the barrier penetration problem, the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined. In the quantum scattering problem, according to the fractional Schrödinger equation, the Green’s function of the Lippmann-Schwinger integral equation is given.
The 2D space-fractional Schrödinger equation in the time-independent and time-dependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.
For many problems in quantum mechanics it proves to be useful to write the Schrödinger equation as an integral equation rather than as a differential equation. In this paper we describe how the transformation of the Schrödinger equation from differential form to integral form may be accomplished. Two methods are described, one appropriate to bound-state theory and one appropriate to scattering theory. It is shown that the bound-state wave function is also a solution of the integral equation (Lippman-Schwinger equation) appropriate to scattering theory.
In this paper, we consider two types of the scattering problems (relativistic case), namely, the stationary scattering problem, where the distance rrr tends to infinity, and the dynamical scattering problem, where the time ttt tends to infinity. Using our results on Lippmann-Schwinger equation in the relativistic case, we found the connection between the stationary scattering problem (the scattering amplitude) and the dynamical scattering problem (the scattering operator). This result is the quantum mechanical analog of the ergodic formulas in the classical mechanics.