Semiclassical study of particle motion in two-dimensional and three-dimensional elliptical boxes. I

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Abstract
The authors compare three problems of quantum mechanics (or more generally of wave mechanics) which reduce to the same problem in classical mechanics and which can also be treated semiclassically by the Einstein, Brillouin and Keller (1958) semiquantisation rules. The free particle motion in an ellipsoidal oblate or prolate cavity (a deformed nucleus) is compared to that in a plane elliptical billiard box. Separation of variables is performed in appropriate coordinate systems. The presence of a separatrix in phase space is exhibited, which is connected to a potential barrier that is different for each problem. The uniform approximation is used to calculate WKB phase rules appropriate to each symmetry. An important difference results between the prolate and the oblate systems.