Quantum mechanics on a Möbius ring: Energy levels, symmetry, optical transitions, and level splitting in a magnetic field

Abstract

We investigate the quantum mechanical energy levels of an electron constrained to motion on a nanoscale M\"obius ring by solving the Schr\"odinger equation on the curved surface. The dimensions of the ring in terms of the lateral and transverse parameters {$u,v$} for the M\"obius ring allow us to identify the quantum numbers for the levels by $({n}_{u},{n}_{v})$. We show that the energy levels can still be labeled using the quantum numbers of the cylindrical ring of the same dimensions. While the Hamiltonian has invariance under parity in parameter space, the rotational symmetry about any axis in configuration space is lost, so that the double degeneracy of energy levels for azimuthal quantum number ${n}_{u}\ensuremath{\ge}1$, that exists in cylindrical rings, is lifted by a small amount in the M\"obius ring. The pattern of level splitting has been identified in terms of the number of twists $\ensuremath{\sigma}$ to be $2{n}_{u}=s\ensuremath{\sigma}$ where $s$ is an integer. The scaling properties of the energy levels with respect to the dimensions of the ring are derived; using these properties, our numerical results which are given for a specific geometry can be extended to rings of other commensurate dimensions. The absence of rotational invariance for the M\"obius ring manifests itself through the orbital angular momentum ${L}_{z}$ not commuting with the Hamiltonian. Its expectation values are found to have nearly integral as well as half-integral values of $\ensuremath{\hbar}$, and its variances are small. The energy levels with half-integral azimuthal quantum numbers (${n}_{u}$) are also close to the approximate formula for the equivalent cylindrical ring, provided such half-integral quantum numbers are allowed for the cylindrical geometry. The Zeeman splitting of the energy levels in an external magnetic field is displayed, together with wave functions at a level anticrossing. The optical transitions between electronic states on the M\"obius ring are obtained, and a table of oscillator strengths is provided. The results for energy levels for rings with multiple twists are presented. In view of recent technological advances in the production of graphene sheets, we may anticipate the making of such twisted rings with graphene strips of finite width. Graphene strips of finite width have an open band gap at the $K$ points in the Brillouin zone, so that a nonrelativistic treatment with a small effective mass is appropriate. For M\"obius rings of graphene, our results would be directly relevant, and we may anticipate their experimental verification in the near future.