Electrostatic-field-induced dynamics in an ultrathin quantum well

TL;DR

This paper derives an exact propagator for a quantum particle in a one-dimensional delta potential under an electrostatic field, and simplifies key components for better physical interpretation and numerical evaluation.

Contribution

It provides an exact expression for the propagator and a simplified approximation for the ionization-recombination component in an electrostatic field with a delta potential.

Findings

Exact propagator derived for the system

Simplified approximation for key propagator component

Enhanced interpretability and numerical accessibility

Abstract

We consider the time evolution of a particle subjected to both a uniform electrostatic field F and a one-dimensional delta-function potential well. We derive the propagator K_F(x,t|x',0) of this system, directly leading to the wavefunction psi_F(x,t), in which its essential ingredient K_F(0,t|0,0), accounting for the ionization-recombination in the bound-continuum transition, is exactly expressed in terms of the multiple hypergeometric functions F(z_1,z_2,...,z_n). And then we obtain the ingredient K_F(0,t|0,0) in an appropriate approximation scheme, expressed in terms of the generalized hypergeometric functions p_F_q(z) being much more transparent to physically interpret and much more accessible in their numerical evaluation than the functions F(z_1,z_2,...,z_n).