Exact Propagator of a Two Dimensional Anisotropic Harmonic Oscillator in the Presence of a Magnetic Field

Abstract

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.