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American Journal of Physics

An alternative approach for the propagator of a charged harmonic oscillator in a magnetic field

Carlos Farina, M. Maneschy, Clifford Neves

Published Jul 1, 1993• 3 citations
Publisher

Abstract

An alternative way of computing the propagator of a nonrelativistic (spinless) charged harmonic oscillator in a constant magnetic field through an appropriate variable transformation directly in the Schrödinger equation is proposed. Due to the axial symmetry exhibited by this problem, polar coordinates are used. Also given is a physical interpretation for these results on the light of both classical and quantum mechanics.

DOI
10.1119/1.17226

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An alternative way of computing the propagator of a nonrelativistic (spinless) charged harmonic oscillator in a constant magnetic field through an appropriate variable transformation directly in the Schrödinger equation is proposed. Due to the axial symmetry exhibited by this problem, polar coordinates are used. Also given is a physical interpretation for these results on the light of both classical and quantum mechanics.
In this paper we solve exactly the problem of the spectrum and Feynman propagator of a charged particle submitted to both an anharmonic oscillator in the plane and a constant and homogeneous magnetic field of arbitrary strength aligned with the perpendicular direction to the plane. As we shall see in the beginning of the letter, the Hamiltonian, being a quadratic form, is easily diagonalizable and the Classical Action can be used to construct the exact Feynman Propagator using the Stationary Phase Approximation. The result is useful for the treatment of quasi two dimensional samples in the field of magnetic effects in nano-structures and quantum optics. The presented solution, after minor extensions, can also be used for motion in three dimensions, and in fact it has been used for years in such cases. Also it can be used as a good exercise of a Feynman Path Integral that can be calculated easily with just the help of the Classical Action.