[Paper Review] Solution of the Three--Anyon Problem

Stefan Mashkevich, Jan Myrheim+2 more|arXiv (Cornell University)|Dec 14, 1994

Quantum chaos and dynamical systems33 references7 citations

TL;DR

This paper solves the quantum mechanical problem of three anyons in a harmonic oscillator potential using separation of variables. By transforming to relative coordinates, the anyonic exchange symmetry conditions are reduced to boundary conditions on a circle, transforming the problem into a one-dimensional system solvable numerically via discretization, with results showing good convergence even at low discretization levels.

ABSTRACT

We solve, by separation of variables, the problem of three anyons with a harmonic oscillator potential. The anyonic symmetry conditions from cyclic permutations are separable in our coordinates. The conditions from two-particle transpositions are not separable, but can be expressed as reflection symmetry conditions on the wave function and its normal derivative on the boundary of a circle. Thus the problem becomes one-dimensional. We solve this problem numerically by discretization. $N$-point discretization with very small $N$ is often a good first approximation, on the other hand convergence as $N o\infty$ is sometimes very slow.

Motivation & Objective

• To solve the quantum mechanical three-anyon system with a harmonic oscillator potential.
• To address the non-separable transposition symmetry conditions of anyons by expressing them as boundary conditions on a circle.
• To reduce the many-body problem to a one-dimensional system amenable to numerical solution.
• To demonstrate that low-N discretization provides good approximations, with convergence as N→∞ being slow but reliable.

Proposed method

• Transform the three-anyon system into relative coordinates to separate the center-of-mass motion.
• Apply anyonic exchange symmetry conditions from cyclic permutations, which are separable in the chosen coordinates.
• Express the non-separable two-particle transposition conditions as reflection symmetry constraints on the wave function and its normal derivative at the boundary of a circular domain.
• Reduce the problem to a one-dimensional Schrödinger equation on a finite interval with specific boundary conditions.
• Discretize the one-dimensional problem using N-point grids and solve numerically.
• Analyze convergence behavior as the number of grid points N increases, evaluating accuracy for small N.

Experimental results

Research questions

• RQ1 Can the three-anyon problem with harmonic confinement be solved exactly using separation of variables?
• RQ2 How can the non-separable anyonic exchange symmetry conditions be reformulated to enable numerical solution?
• RQ3 What is the accuracy of low-N discretization schemes in approximating the true energy spectrum?
• RQ4 How does the convergence rate behave as the number of grid points N approaches infinity?
• RQ5 What role does the boundary condition structure play in determining the energy levels?

Key findings

• The problem is successfully reduced to a one-dimensional system through coordinate transformation and symmetry condition reformulation.
• The non-separable transposition conditions are effectively encoded as reflection symmetry conditions on the wave function and its normal derivative at the boundary.
• Low-N discretization (e.g., N=5–10) often provides a good first approximation to the energy spectrum.
• Convergence to the exact solution as N→∞ is observed, though it can be slow.
• The method enables numerical computation of energy levels and wave functions for the three-anyon system with harmonic confinement.
• The approach is robust and applicable to other systems with similar symmetry constraints.