Behaviour of three charged particles on a plane under perpendicular magnetic field

TL;DR

This paper analyzes the behavior of three identical charged particles on a plane under a perpendicular magnetic field, revealing two solution sets related to anyons and Landau levels, and connecting to quantum Hall states.

Contribution

It introduces a novel solution framework for three charged particles in a magnetic field using Heun equations, linking anyonic states to quantum Hall wave functions.

Findings

Identified two sets of eigenstates: anyonic and Landau-like.

Connected Laughlin wave functions as special cases of the solutions.

Provided explicit eigenstates in the strong magnetic field limit.

Abstract

We consider the problem of three identical charged particles on a plane under a perpendicular magnetic field and interacting through Coulomb repulsion. This problem is treated within Taut's framework, in the limit of vanishing center of mass vector $\vec{R} \to \vec{0}$, which corresponds to the strong magnetic field limit, occuring for example in the Fractional Quantum Hall Effect. Using the solutions of the biconfluent Heun equation, we compute the eigenstates and show that there is two sets of solutions. The first one corresponds to a system of three independent anyons which have their angular momenta fixed by the value of the magnetic field and specified by a dimensionless parameter $C \simeq \frac{l_B}{l_0}$, the ratio of $l_B$, the magnetic length, over $l_0$, the Bohr radius. This anyonic character, consistent with quantum mechanics of identical particles in two dimensions, is induced by competing physical forces. The second one corresponds to the case of the Landau problem when $C \to 0$. Finally we compare these states with the quantum Hall states and find that the Laughlin wave functions are special cases of our solutions under certains conditions.