The energy levels and the corresponding normalized wave functions for a model of a compressed atom

Abstract

In the model of a compressed atom (or ion) considered in the present paper the boundary condition associated with the corresponding uncompressed atom, i.e., the condition that the radial wave function must vanish at r=∞, is replaced by the boundary condition that the radial wave function must have a node at the finite distance r=a. The treatment of the problem of obtaining the energy shift due to the compression is based on the phase-integral method developed by Fröman and Fröman, an essential feature of which is that one can use exact formulas in the calculations and make all approximations in the final stage. The treatment of the problem of obtaining the relative change of the wave function due to the compression is based on the rigorous evaluation of the normalization integral developed by Furry [Phys. Rev. 71, 360 (1947)] and Yngve [J. Math. Phys. 13, 324 (1972)], in which one also uses exact formulas in the calculations and makes all approximations in the final stage. Since compression of an atom gives rise to very subtle effects, rigorous methods are indispensible for obtaining accurate and reliable analytical final formulas. As an application, the resulting general formulas are particularized to the case of a hydrogenic atom, and a numerical illustration of the accuracy of the formulas is given.