Energy variational principle for systems in magnetic fields: Application to the anisotropic harmonic oscillator

Abstract

A method is developed for improving the trial wave function for a particle in a magnetic field when the Rayleigh-Ritz energy variational principle is used. The energy operator depends on the vector potential, which must be chosen in some gauge. The expectation value of the energy with respect to an arbitrary trial wave function depends on the gauge of the vector potential. When the trial wave function is used to calculate the current, the equation of continuity for charge conservation is not, in general, satisfied. The arbitrary trial wave function can be multiplied by a phase factor which depends on the spatial coordinates. When the energy expectation value is minimized with respect to the phase function, the equation for charge conservation is obtained. This equation can be solved for the phase function, and a solution used in the energy expectation value to obtain a lower energy which is also independent of the choice of the gauge of the vector potential. The method is illustrated by applying it to an anisotropic harmonic oscillator in a constant magnetic field.