Partícula confinada en una cavidad esferoidal prolata. Algunos efectos de la pérdida de simetría esférica.

Abstract

The problem being studied is that of a particle moving unimpeded that is confined within a spheroidal box. Using the software Mathematica the spheroidal functions are plotted for six physical states. The energy levels below 8.5 rydbergs are calculated when the inverse of the excentricity $\xi o $ is taken to be $\xi o=3$ and $\xi o=30$. A comparison is drawn between the energy spectrum of a particle confined within a spherical box of the same volume, and it is found that every level of this system is split in $% l+1 $ levels when its spherical symmetry is shifted to a prolate symmetry. The matrix elements for dipolar interaction are studied and it is shown that the light that is either emitted or absorbed is circularly polarized. The electric dipole moment is calculated for the ground state and for three further excited states. Einstein's $A_{ij}$ coeficients are calculated for six allowed transitions. The simplicity of the computer work being done allows us to suggest that this material is didactically useful to understand key aspects of what happens within a quantum system in the event that its symmetry is modified. Activities are suggested that could be performed by undergraduate students.