Asymptotic solution of the Schrodinger equation for the elliptic wire in the magnetic field

TL;DR

This paper derives an asymptotic solution for the Schrödinger equation in an elliptic wire under a magnetic field, classifying electron states and analyzing energy spectra and probability distributions.

Contribution

It provides a novel asymptotic analytical approach for solving the Schrödinger equation in elliptic geometries with magnetic fields, including classification of quantum states.

Findings

Classification of electron states into Boundary, Ring, Hyperbolic Caustic, and Harmonic Oscillator modes.

Derivation of energy spectra and probability distributions for excited hole states in Bi wires.

Identification of Whispering Gallery and Jumping Ball modes in the system.

Abstract

The asymptotic solution of the Schrodinger equation with non-separable variables is obtained for a particle confined to an infinite elliptic cylinder potential well under applied uniform longitudinal magnetic field. Using standard-problem method, dimension quantized eigenvalues have been calculated when the magnetic length is large enough in comparison with the half of the distance between the boundary ellipse focuses. In semi-classical approximation, the confined electron (hole) states are divided into the Boundary States (BS), Ring States (RS), Hyperbolic Caustic States (HCS), and Harmonic Oscillator States (HOS). For large angular momentum quantum numbers and small radial quantum numbers, the BS and RS are grouped into the Whispering Gallery mode. They associate with particles moving along the wire cross section boundary. The motion is limited from the wire core by the elliptic caustic. Consisting of the HCS and HOS, the Jumping Ball modes correspond to the states of particle moving along a wire diameter when the angular momentum quantum number is much less than the radial quantum number. In this case, the motion is restricted by the hyperbolic caustics and two boundary ellipse arcs. For excited hole states in Bi wire, the energy spectrum and space probability distribution are analyzed.