Symmetry analysis for a charged particle in a certain varying magnetic field

TL;DR

This paper investigates the symmetries of a charged particle's motion in a varying magnetic field using Lie group analysis and spectrum generating algebra, revealing specific symmetry structures and energy level properties.

Contribution

It applies Lie symmetry analysis and algebraic methods to analyze classical and quantum dynamics in a non-uniform magnetic field, providing new insights into their symmetry properties.

Findings

Identified Lie point symmetries of the classical equations of motion.

Used $su(1,1)$ algebra to find quantum energy levels.

Discovered specific vector field terms and non-closure of Lie algebra for certain eigenvalues.

Abstract

We analyze the classical equations of motion for a particle moving in the presence of a static magnetic field applied in the $ z $ direction, which varies as $ {1\over{x^2}} $. We find the symmetries through Lie's method of group analysis. In the corresponding quantum mechanical case, the method of spectrum generating $su(1,1)$ algebra is used to find the energy levels for the Schroedinger equation without explicitly solving the equation. The Lie point symmetries are enumerated. We also find that for specific eigenvalues the vector field contains $ {1\over{x}} {{\p}\over{\p x}}$ and $ {1\over {x^2}} {{\p}\over{\p {x}}}$ type of terms and a finite Lie product of the generators do not close.