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 $\cosh(x)\overset{ x \to i x}{ \longrightarrow}\cos(x)$.
Categories: quant-ph