EIGENVALUE SPECTRUM FOR A SINGLE PARTICLE IN A SPHEROIDAL CAVITY: A SEMICLASSICAL APPROACH

Abstract

Following the semiclassical formalism of Strutinsky et al.,11 we have obtained the complete eigenvalue spectrum for a particle enclosed in an infinitely high spheroidal cavity. Our spheroidal trace formula also reproduces the results of a spherical billiard in the limit η → 1.0. Inclusion of repetition of each family of the orbits with reference to the largest one significantly improves the eigenvalues of the sphere and an exact comparison with the quantum mechanical results is observed up to the second decimal place for kR0 ≥ 7. The contributions of the equatorial, planar (in the axis of symmetry plane) and non-planar (three-dimensional) orbits are obtained from the same trace formula by using the appropriate conditions. The resulting eigenvalues compare very well with the quantum mechanical eigenvalues at normal deformation. It is interesting that the partial sum of equatorial orbits leads to eigenvalues with maximum angular momentum projection, while the summing of planar orbits leads to eigenvalues with Lz = 0 except for L = 1. The remaining quantum mechanical eigenvalues are observed to arise from the three-dimensional (3D) orbits. Very few spurious eigenvalues arise in these partial sums. This result establishes the important role of 3D orbits even at normal deformations.