An algebraic approach to a charged particle in an uniform magnetic field

TL;DR

This paper explores the quantum states of a charged particle in a uniform magnetic field using algebraic methods, revealing connections between different gauge choices and coherent states.

Contribution

It introduces an algebraic framework for analyzing the Landau and symmetric gauges, linking harmonic oscillator and $SU(1,1)$ coherent states through group contraction.

Findings

Eigenfunctions in Landau gauge are harmonic oscillator number coherent states.

Explicit form of $SU(1,1)$ Perelomov number coherent states in symmetric gauge.

Demonstrates the contraction from $SU(1,1)$ to Heisenberg-Weyl group relating different coherent states.

Abstract

We study the problem of a charged particle in a uniform magnetic field with two different gauges, known as Landau and symmetric gauges. By using a similarity transformation in terms of the displacement operator we show that, for the Landau gauge, the eigenfunctions for this problem are the harmonic oscillator number coherent states. In the symmetric gauge, we calculate the $SU(1,1)$ Perelomov number coherent states for this problem in cylindrical coordinates in a closed form. Finally, we show that these Perelomov number coherent states are related to the harmonic oscillator number coherent states by the contraction of the $SU(1,1)$ group to the Heisenberg-Weyl group.