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Two Dimensional Wave Packages Scattering on Step and Rectangular Potentials

Georgii Nikolaevich Krylov

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

Wave packages reflection on step-like potentials can lead to significant deviation of the mean wave vector in respect to equal angles law due to dispersion of scattering on the region boundary. The calculated cases are step and rectangular potentials for two-dimensional Schr¨odinger equation and Dirac equation for graphene-like charge carriers quasi-particles.

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Wave packages reflection on step-like potentials can lead to significant deviation of the mean wave vector in respect to equal angles law due to dispersion of scattering on the region boundary. The calculated cases are step and rectangular potentials for two-dimensional Schr¨odinger equation and Dirac equation for graphene-like charge carriers quasi-particles.
One-dimensional wave package propagation through the potential step is considered. Several approaches are tested for the ability to obtain transmission coefficients. Consideration includes direct numeric solution of the time-dependent Schrödinger Equation with the step potential, usage of the spectral decomposition of the propagator. An asymptotic formula is proposed for calculation of the transmission coefficient, which produces accurate results and is much easier to apply then the others considered approaches.
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A propagator for the one-dimensional time-dependent Schr¨odinger equation with an asymmetric rectangular potential is obtained, by using the multiple-scattering theory. It allows the consideration of the reflection and transmission processes as the scattering of a particle at the potential jump (in contrast to the conventional wave-like picture) and the account for the non-classical counterintuitive contribution of the backward-moving component of the wave packet attributed to a particle. This propagator completely resolves the corresponding time-dependent Schr¨odinger equation (defines the wave function w(x, t)) and allows the consideration of the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of the time. These results are related to fundamental issues such as measuring the time in quantum mechanics (tunneling time, time of arrival, dwell time). For the imaginary time, which represents an inverse temperature (t → −ih/kBT), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to the action of a rectangular potential. This density matrix provides information about particles’ density in the different spatial areas relative to the potential location and on the quantum coherence of different particle spatial states. If one passes to the imaginary time (t → −it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals of elementary functions and thus allow a numerical visualization of the probability density |w(x, t)|2, the density matrix, and the solution of the diffusion-like equation. The results obtained may also be applied to spintronics due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
Under strong coupling conditions,the non-relativistic motion of particles and spin zero or half-integer the nature of relativistic particles in the system with the Schrdinger equation and Dirac equation can respectively describe.A new ring-shaped harmonic oscillator potential is proposed in this paper.The exact bound solution of Dirac equation and Schrdinger equation for the above potential is obtained under the condition of equal scalar potentials and vector potentials.They can be separated into an corresponding to the wave function equations of angular and radial wave function equation with the method of separation variables.The normalized angle wave function and radial wave function were expressed respectively in terms of the generalized associated-Legendre function and the confluent hyper-geometric function are presented,that the normalized radial wave function,and then by the radial bound state wave function are satisfied with the boundary conditions,obtains the exact bound state energy spectrum equations.The exact energy spectrum equations are obtained and meanwhile,proper discussion and some important conclusions are presented.