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Communications in Theoretical Physics

Numerical Solution of the Four-Anyon Problem

Ruan Wenying

Published Jan 30, 1998• 0 citations
Publisher

Abstract

By making use of the Talmi–Moshinsky transformation bracket, the Schrodinger equation for four anyons in configuration space has been solved variationally to obtain the eigenenergies and eigenfunctions. The low-lying states have been presented as functions of the statistical parameter. We illustrated how the bosonic states evolve continuously into the fermionic states.

DOI
10.1088/0253-6102/29/1/19

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By making use of the Talmi–Moshinsky transformation bracket, the Schrodinger equation for four anyons in configuration space has been solved variationally to obtain the eigenenergies and eigenfunctions. The low-lying states have been presented as functions of the statistical parameter. We illustrated how the bosonic states evolve continuously into the fermionic states.
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In most cases solving a quantum mechanical problem exactly is not always possible. In such a case, an analytic solution to the Schrödinger equation is appropriate. This results to numerical solutions that may not necessarily be the best approach. Interest sometimes may not necessarily be in the detailed eigenfunctions, but rather in the energy levels and the qualitative features of the eigenfunctions. So, the numerical solutions are usually less intuitively reasonable. It is possible to find the values of the energy levels to be slightly sensitive to the deviation of the wave function from its true form. So, the expectation value of the energy for an approximate wave function can be a very good estimate of the corresponding energy eigenvalue. Using an approximate wave function dependent on some set of variational parameters and minimizing its energy by these variational parameters yield such energy estimates. This technique is the so-called variational method due to this minimization procedure.