An asymptotic solution of the Schrödinger equation for the elliptic wire in the magnetic field

DOI10.1088/1751-8113/41/39/395304zbMATH Open1192.81119arXiv0803.0942OpenAlexW2049961007MaRDI QIDQ3531184

Publication date: 21 October 2008

Published in: Journal of Physics A: Mathematical and Theoretical (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/0803.0942

Mathematics Subject Classification ID

Quantum computation (81P68) Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Asymptotic distributions of eigenvalues in context of PDEs (35P20) Schrödinger operator, Schrödinger equation (35J10) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20)