An exact solution of the time-dependent Schr\"odinger equation with a rectangular potential for real and imaginary time

TL;DR

This paper derives an exact propagator for the time-dependent Schrödinger equation with an asymmetric rectangular potential, enabling detailed analysis of quantum scattering, tunneling, and thermal effects, with applications in spintronics.

Contribution

It provides a novel exact solution for the propagator in asymmetric rectangular potentials, incorporating reflection, transmission, and backward wave components, applicable to real and imaginary time.

Findings

Exact propagator expressed as elementary integrals

Able to visualize wave function and density matrix evolution

Applicable to spintronics and layered nanostructures

Abstract

A propagator for the one dimensional time-dependent Schr\"odinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schr\"odinger equation (defines the wave function ${\psi}(x,t)$) and allows for considering 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 time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature $(t\rightarrow-i{\hbar}/k_{B}T)$, the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. If one shifts to imaginary time $(t\rightarrow-it)$, 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 from elementary functions and thus allow for a numerical visualization of the probability density $|\psi(x,t)|^2$, the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.