An analytic study of the ionization from an ultrathin quantum well in a weak electrostatic field

TL;DR

This paper provides explicit analytical expressions for the time evolution and ionization probability of a particle in a one-dimensional delta-function potential under a weak electrostatic field, including a new integral identity of the Airy function.

Contribution

It derives a closed-form approximation for the wavefunction at the origin and ionization probability, validated against exact solutions, and introduces a new integral identity of the Airy function.

Findings

Approximate wavefunction at the origin closely matches the exact solution.

Ionization probability derived analytically agrees with numerical results.

New integral identity of the Airy function discovered.

Abstract

We consider the time evolution of a particle bound by an attractive one-dimensional delta-function potential (at x = 0) when a uniform electrostatic field (F) is applied. We explore explicit expressions for the time-dependent wavefunction \psi_F(x,t) and the ionization probability {\mathcal{P}}(t), respectively, in the weak-field limit. In doing so, \psi_F(0,t) is a key element to their evaluation. We obtain a closed expression for \psi_F(0,t) which is an excellent approximation of the exact result being a numerical solution of the Lippmann-Schwinger integral equation. The resulting probability density |\psi_F(0,t)|^2, as a simple alternative to {\mathcal{P}}(t), is also in good agreement to its counterpart from the exact one. In doing this, we also find a new and useful integral identity of the Airy function.