Schrödinger equation for convex plane polygons: A tiling method for the derivation of eigenvalues and eigenfunctions
- 1 September 1991
- journal article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 32 (9) , 2442-2449
- https://doi.org/10.1063/1.529172
Abstract
Motivated by a recently advanced conjecture on the ergodic properties of Quantum Systems, the problem of solving the Schrödinger equation for a free particle in a plane polygonal enclosure is revisited. It will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves, without making use of advanced group-theoretical techniques. It turns out, inter alia, that these polygons, considered as classical billiards, are all and only those which are completely integrable in the sense of Arnold’s theorem.Keywords
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