Quantum mechanics of a particle in a ballistic Corbino ring with finite-potential barriers

Summary

We present in this paper a quantum-mechanical problem of a particle in a Corbino ring (i.e. a ring-shaped quantum wire inR ²) with finite-potential barriers, and consider i) the mathematical aspects associated with a quantum mechanics of the particle for a Corbino geometry and ii) the confinement (or escape) problem of the particle in the ring. It is shown that by deforming a Hilbert space, the Schrödinger equation for the system can be reduced to a radial Schrödinger equation expressed in terms of momentum and configuration operators in a new physical Hilbert space. Accordingly, one can treat the problem as an ordinary one-dimensional quantum-mechanical problem and so the confinement problem of the particle in a simple manner. The radial Schrödinger equation contains an effective confinement potential, which the particle actually experiences. The arguments are given in detail to the constituent of this Schrödinger equation including a singular term. Evaluating the singularity and solving the Schrödinger equation equation, we discuss the feasibility of an escape of the particle from the ring. We find that the tunnelling (or quantal leak) of the particle from the ring could occur due to the centrifugal force and the quantum effect arising from the distortion of the confinement potential originated in a curved geometry. The criterion of the tunnelling for the ballistic particle through a part of the deformed potential in given in terms of the radius and thickness of the ring. By applying the KWB method, we also derived a formula for a transmission probability (or a Gamov penetration factor) of the particle’s tunnelling phenomenon.