Solution of Time-Independent Schr?dinger Equation for a Two-Dimensional Quantum Harmonic Oscillator Using He's Homotopy Perturbation Method

In this paper, time-independent Schr\"{o}dinger equation for a charged particle, in the presence of electric potential and vector potential, has been solved using He's Homotopy Perturbation Method (HPM). HPM is one of the newest analytical methods to solve linear and nonlinear differential equations. In contrast to the traditional perturbation methods, the Homotopy method does not require a small parameter in the equation. In this method, according to the homotopy technique, a Homotopy with an embedding parameter $\delta \in \lbrack 0,1]$ is constructed, and the embedding parameter is considered as a small parameter. Using cylindrical coordinates, it has been found that the z-equation of the charged particle is a one-dimensional harmonic oscillator and the r equation is actually a two-dimensional harmonic oscillator. The obtained results show the evidence of simplicity, usefulness, and effectiveness of the HPM for obtaining approximate analytical solutions for the time-independent Schr\"{o}dinger equation for a charged particle in parallel electric and magnetic fields.