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Describing Two Charged Particles In A Magnetic Field Using Trajectories And Wave Equations Through Factorization

Student Name: Hall, Adam
Advisor Name: Van Huele, Jean-Francois
Approval Date: 4/1/2009
Report Type: Senior Thesis
Title: Describing Two Charged Particles In A Magnetic Field Using Trajectories And Wave Equations Through Factorization
Abstract: We examine the nonrelativistic classical equations of motion of two charged massive particles in a static homogeneous magnetic eld. We discuss criteria for when the classical nonrelativistic radiationless approximation is valid for this two-particle system. We focus on the motion in the plane perpendicular to the direction of the magnetic eld. We determine conditions of boundedness in this plane for both the center-of-mass vector and the relative-position vector that describe the two-particle system. We then examine a spinor equation that describes two nonrelativistic quantum particles in a homogeneous magnetic eld. By treating some of the terms of the Hamiltonian as perturbations, we obtain analytic expressions for the energy levels of the two-particle system. We apply these analytic expression to predict the energy levels for neutral two-body systems such as hydrogen and positronium, and for the positive helium ion and any hydrogen-like ions. Finally, we explore a matrix factorization technique to derive nonrelativistic quantum wave equations which may incorporate spin, and relativistic quantum wave equations which may incorporate anti-particle wave components. From our investigation, we postulate the necessary conditions to obtain Schrodinger wave equations, Pauli wave equations, Klein-Gordon wave equations, and Dirac wave equations.
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