Schrödinger equation is $\mathcal{R}$-separable in toroidal coordinates

TL;DR

The paper tackles solving the Schrödinger equation in toroidal geometries where conventional separation fails. It applies irregular $\\mathcal{R}$-separation in Moon and Spencer's toroidal coordinates and introduces a toroidal–poloidal coordinate system to recover periodicity, enabling analytic progress otherwise deemed impossible. The main contributions are exact eigenfunctions for a free particle and for a toroidal well under external potentials, expressed in terms of $J_m$, $H_m^{(1)}$, and Heun functions, along with a Green function and plane-wave expansion that facilitate scattering analyses. A key finding is the emergence of fractional angular momentum due to the torus topology, with potential implications for toroidal quantum wells and related devices, and the framework paves the way for further analytic treatment of toroidal geometries in quantum mechanics.

Abstract

We present, for the first time, exact solutions for the Schrödinger equation in Moon and Spencer's toroidal coordinates, and in the electromagnetic toroidal--poloidal coordinate systems. Curiously, both systems present a fractional angular momentum, because of the torus's hole. We achieve these novel solutions using the irregular $\mathcal{R}$-separation of variables, an unexplored approach in Physics, which results in a wavefunction with fractional angular momentum eigenvalues. Numerous solutions for the Schrödinger equation in a variety of external potentials are shown, including an external magnetic field. A plane-wave expansion and a Green function are also presented, setting the stage for future progress in this area.

Figures (4)

• Figure 1: Level surfaces of the toroidal/poloidal coordinate system. We see a yellow torus, as a surface of constant $w$, cyan cones, of constant $u$, and magenta disks, setting $v$ constant.
• Figure 2: Cross section of a torus with minor radius $a$, major radius $R$ for a fixed $v$, where the plane $\rho$ is defined by $z = 0$ and $\rho = \sqrt{x^2 + y^2}$.
• Figure 3: Probability densities $\sigma |\psi_{0}(w,u,v)|^2$ and $\sigma |\psi_{2}(w,u,v)|^2$, $k_{0,1} \approx 2.4048$, the first zero of $J_0(x)$, where $\sigma=\sqrt{g}|_{w = w'}$ given by \ref{['det-g']}. Observe that the probability densities vanish at the toroidal shell. Here we used $R=1, m^*=1/2$ and $\hbar=1$.
• Figure 4: Probability densities of $\sigma |\psi_{1}(w,u,v)|^2$ and $\sigma |\psi_{3}(w,u,v)|^2$, using $k = k_{4,0}$ and $k = k_{5,0}$, the fourth and fifth zeros of $J_0(x)$, where $\sigma$ is the surface element and we used the same values of the previous figure: $R=1, m^*=1/2$ and $\hbar=1$. We observe that there is a region where the probability densities vanish (purple tube) inside the well in both of them. Maxima takes place at red regions.