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American Journal of Modern Physics

An Algebraic Operator Approach to Aharonov-Bohm Effect

Farrin Payandeh

Published Jan 1, 2015• 0 citations
Publisher

Abstract

A new approach based on algebraic quantum operator, is pursued in order to investigate the Aharonov-Bohm effect. Introducing a SU(2) dynamical invariance algebra, the discrete spectrum and the energy level of the quantum Aharonov-Bohm effect is obtained. This alternative method will help undergraduate students to broader their knowledge about this interesting quantum phenomenon.

DOI
10.11648/j.ajmp.20150402.12

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A new approach based on algebraic quantum operator, is pursued in order to investigate the Aharonov-Bohm effect. Introducing a SU(2) dynamical invariance algebra, the discrete spectrum and the energy level of the quantum Aharonov-Bohm effect is obtained. This alternative method will help undergraduate students to broader their knowledge about this interesting quantum phenomenon.
Aharonov–Bohm effect is a quantum mechanical phenomenon that attracted the attention of many physicists and mathematicians since the publication of the seminal paper of Aharonov and Bohm [1] in 1959. We consider different types of Aharonov–Bohm effects such as the magnetic AB effect, electric AB effect, combined electromagnetic AB effect, AB effect for the Schrödinger equations with Yang–Mills potentials, and the gravitational analog of AB effect. We shall describe different approaches to prove the AB effect based on the inverse scattering problems, the inverse boundary value problems in the presence of obstacles, spectral asymptotics, and the direct proofs of the AB effect.
The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schrddoto\\ddot{o}ddotodinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schrddoto\\ddot{o}ddotodinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.