Spatially confined simple quantum mechanical systems

A study of two simple quantum mechanical system, i.e., harmonic oscillator (HO ) and hydrogen atom (H ) trapped inside the wells is performed. The influence of the spatial confinement on the energy spectra of both systems is investigated. An analytically solvable model of the well potential is presented in the case of the HO . A central and noncentral localization of HO in the well is discussed. The results are used to explain some reported anomalous experimental data on the porphine IR spectrum. The case of hydrogen atom is studied by solving the problem of the electron in the potential V = Z/R cot r/R. An exact formula for the energy is derived for s states. For others, analytically nonsolvable symmetries, a formula with correct asymptotic behavior is proposed. The results are compared with some numerical calculations for spherical rectangular well and an extension to many-electron atoms is also presented. The effect of breaking degeneracy and of ordering energies of H in a different way than in the case of many-electron systems is broadly discussed. © 1994 John Wiley & Sons, Inc.