Exact solutions of the hydrogen-like atoms confined by a potential step

The derivation of the exact solutions of the hydrogen-like atoms confined by a potential step was analyzed. It was shown that the quantum effects of the doping atom caused by a confining potential were different from a normal free space atom. It was suggested that the asymptotic behaviours for the confluent hypergeometric functions M (a,b,z) and U(a,b,z) at the origin and at infinity should be considered in order to make the radial wave functions regular at the origin and at infinity. It was found that the calculation of the eigenvalues became very complicated since the eigenvalues were closely related to the potential parameter V₀, the radius r₀, and the angular momentum quantum number m or l.