Abstract
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.
Highlights
The dynamics of charged particles in electric and magnetic fields is of both academic and practical interest in physics and engineering
A charged particle in a time-independent homogeneous magnetic field executes a circular motion in the plane perpendicular to the direction of the field
The period of this motion is the inverse of the cyclotron frequency ωc = qB/m, where q is the charge of the particle, m is the mass of the particle, and B is the strength of the magnetic field
Summary
The dynamics of charged particles in electric and magnetic fields is of both academic and practical interest in physics and engineering. Fendrik and Bernath (1989) has solved the Schrodinger equation for the two-dimensional simple harmonic oscillator using elliptic coordinates where it is separable. The eigenstates and the corresponding eigenvalues of a charged particle, in the presence of electric potential and vector potential, are obtained by solving the time-independent Schrodinger equation in cylindrical coordinates. The method, which is a coupling of a homotopy technique and a perturbation technique, deforms continuously to a simple problem which is solved This method, which does not require a small parameter in an equation, in contrast to the traditional perturbation methods, has a significant advantage in that it provides an analytical approximate solution to a wide range of linear and nonlinear problems in applied sciences.
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