Revisiting the generalized pseudospectral method: Energy degeneracies, radial expectation values, and Schwarz‐like inequalities of the shell‐confined atom

Abstract

AbstractWe revisit the generalized pseudospectral (GPS) method to investigate the physical properties of shell‐confined atoms. Based on a general mapping function, the GPS method is developed for obtaining highly accurate bound state energies, wave functions, and radial quantities for the shell‐confined hydrogen atom. Besides the incidental and simultaneous degeneracies in the energy spectrum, we find that energy degeneracy appears very commonly in the shell‐confined system. The contour maps of the eigenenergies for some low‐lying bound states are obtained and the corresponding wave functions are analyzed. Radial expectation values with both positive and negative powers are calculated with high accuracy. By analyzing the Schwarz‐like inequalities, we show that the ground state density function for the outer‐confined hydrogen atom is second‐order monotonic, while for the shell‐confined hydrogen atom the radial densities for all bound states are zeroth‐order monotonic. In the extreme situation when the inner radius coalesces with the outer radius, the electron becomes a classical particle.