Energy spectra of a particle confined in a finite ellipsoidal shaped potential well

Highlights: • A particle confined in a finite strongly prolate ellipsoidal well is studied. • The transcendental equation for energy levels is obtained and eigenfunctions are found. • The formalism is generalized for a different effective-mass in- and outside of the ellipsoid. • The calculated energy levels are in a good agreement with numerical solutions. A charged particle confined in a strongly prolate ellipsoidal shaped finite potential well is studied. In the case when a distance $R$ between foci is large and accordingly $R^{-1}$ is small, the asymptotic solutions of quasiradial and quasiangular equations in prolate spheroidal coordinates are found. We demonstrate that quasiangular wave functions inside and outside of the potential well coincide on the entire surface of strongly prolate ellipsoid if separation parameters are chosen appropriately. This allows us to obtain the transcendental equation for the energy levels by equating the quasiradial wave function and its derivative on the surface of ellipsoid. The obtained equation is solved numerically and algebraically. The calculated energies are in good qualitative and quantitative agreement with the results obtained earlier for the infinitely high ellipsoidal potential well via a numerical solution of the quasiradial and quasiangular equations. An importance of the actual shape of ellipsoidal potential well for calculation of the energy spectrum for the trapped particle is shown. A dependence of the energy spectrum on the effective mass when it is a different constant inside and outside of the ellipsoid is addressed.