Quantum eigenvalues and eigenfunctions of an electron confined between conducting planes

TL;DR

The paper studies an electron confined between two grounded planes separated by $L$, producing a symmetric double-well image-potential that couples a Coulomb-like problem with a particle-in-a-box. The main approach derives a closed-form image potential in terms of the digamma function, $V(x)=\frac{1}{4L}[\psi(a)+\psi(1-a)+2\gamma]$ with $a=x/L$, and solves the resulting Schrödinger equation using a discrete variable representation on a Chebyshev grid. Key findings include PIB-like energy scaling $E_N(L)\approx \tfrac{1}{2}(\pi N/L)^2$ for small $L$ and bound-image energies $-\tfrac{1}{32n^2}$ for large $L$, plus a tunneling-induced level splitting $\Delta E(L)$ that agrees with analytical forms derived from the ground-state wavefunction of a single plane. The work demonstrates an accessible computational framework (in Julia) for exploring confined quantum systems with image potentials and has potential relevance to bilayer graphene and quantum-dot setups.

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 Schrödinger'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.

Figures (7)

• Figure 1: Comparison of Eq. (6) with and without $2\gamma$ vs Eq. (7). L = 1.0
• Figure 2: Log(E) verses Log(N) for L=1.0, M=100
• Figure 3: Log(quantum defect) vs log(N) for N=45:65, L=1.0, M=100
• Figure 4: Energy of the first 10 eigenvalues as function of distance L
• Figure 5: Energy splitting of first eigenvalue pair as function of distance L