The Quantum Mechanical Problem of a Particle on a Ring with Delta Well

TL;DR

This paper provides analytical solutions for a quantum particle on a ring with an attractive delta potential, revealing bound and unbounded states through transcendental equations involving hyperbolic and trigonometric functions.

Contribution

It derives closed-form solutions for the bound and unbounded states of a particle on a ring with a delta potential, including the transcendental equations governing their parameters.

Findings

Existence of a single bound state with hyperbolic cosine wavefunction.

Infinite set of unbounded solutions related to the bound state via complex substitution.

Explicit form of energy eigenvalues and wavefunctions for the system.

Abstract

The problem of a spin-free electron with mass $m$, charge $e$ confined onto a ring of radius $R_0$ and with an attractive Dirac delta potential with scaling factor (depth) $\kappa$ in non-relativistic theory has closed form analytical solutions. The single bound state function is of the form of a hyperbolic cosine that however contains a parameter $d>0$ which is the single positive real solution of the transcendental equation $\coth(d) = \lambda d$ for non zero real $\lambda=\frac{2}{\pi\kappa}$. The energy eigenvalue of the bound state $\varepsilon=-\frac{d^2}{2\pi^2}\approx \frac{q e m R_0}{2 \hbar^2}$. In addition a discretly infinite set of unbounded solutions exists, formally these solutions are obtained from the terms for the bound solution by substituting $d \to i d $ yielding $\cot(d) = \lambda d$ as characteristic equation with the corresponding set of solutions $d_k, k\in\mathbb{N}$, the respective state functions can be obtained via $\cosh(x)\overset{x \to i x}{\longrightarrow}\cos(x)$.