Quantum eigenvalues and eigenfunctions of an electron confined between conducting planes

TL;DR

This paper analyzes the quantum behavior of an electron confined between grounded conducting planes, deriving solutions to Schrödinger's equation for the resulting double well potential using spectral methods.

Contribution

It provides a novel derivation of the electrostatic potential from image charges and solves the quantum problem with spectral techniques, connecting limiting cases.

Findings

Derived a compact expression for the electrostatic potential between grounded planes.

Solved Schrödinger's equation for the system using spectral methods.

Identified tunneling level splitting during the transition from large to small plane separation.

Abstract

Two of the most iconic systems of quantum physics are the particle in a box and the Coulomb potential (the third is, of course, the harmonic oscillator). In this expository paper, we consider the quantum solution to the problem of an electron confined between the grounded planes of an infinite capacitor. The potential arises from the image charges that form in the grounded planes, along with the added condition that at x = 0, L, where L is the distance between the planes, the wavefunction must be zero. This effectively couples a hydrogen like system to a particle-in-a-box (PIB) based on L, the distance between the planes. The problem of finding the electrostatic potential of this infinite series of image charges is an old one, going back to at least 1929. Here, we give a short derivation for one of the limiting cases that yields a compact expression and show how the Kellogg infinite summation formula converges to that value. We note here that this potential is a symmetric double well potential, so there will be many familiar properties of its solutions. Then using that potential, we solve Schr\"odinger's equation using a spectral technique. The limiting forms of a particle in a box for small L (and high E), and that of a (degenerate) bound image charge at large L and small energy are recovered. We also discuss the tunneling level splitting that occurs in the transition from the large L to the small L regime.