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The Particle in a Box (and in a Circular Box)

Carl W David

Published Jan 1, 2006• 5 citations
OpenAlex

Abstract

The particle in a box problem in 1 and 2 dimensions is treated both for the Cartesian problem (square, rectangle) but for circular boundary conditions.

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In the present work we introduce a methodology for solving a quantum billiard with Dirichlet boundary conditions. The procedure starts from the exactly known solutions for the particle in a circular disk, which are subsequently radially rescaled in such a way that they obey the new boundary conditions. In this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision. Test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy.