Relation between Poisson and Schr\"odinger equation in one dimension

TL;DR

This paper establishes a simple transformation linking the Poisson and Schr"odinger equations in one dimension, enabling solutions for quantum bound states in systems with delta potentials, such as trapping electrons in ionic crystals.

Contribution

It introduces a novel transformation connecting electrostatics and quantum mechanics for specific potentials, facilitating exact solutions for bound states.

Findings

The analogy applies only to potentials with mixed attractive and repulsive delta functions.

Elementary electrostatic results can be used to solve quantum bound state problems.

Potential to trap a single electron in a one-dimensional ionic crystal.

Abstract

The relation between the Poisson and Schr\"odinger equation in one dimension is obtained through a simple transformation. It is pointed out that this analogy between both equations can be only applied for potentials that involve a combination of attractive and repulsive delta function potentials. This relationship enables us to use elementary electrostatic results to find the exact solution of the associated quantum bound state problem in one dimension. Particularly, the result shows that it is possible to trap a single electron in a one dimensional ionic crystal.