Bound eigenstates for the superposition of the Coulomb and the Yukawa potentials.

Abstract

The eigenvalue problem for two particles interacting through the potential being the superposition of the attractive Coulomb potential (-A/r) and the Yukawa potential B exp(-Cr)/r of arbitrary strength B and screening parameter C is solved by variational means. The energy levels ${E}_{\mathrm{nl}}$ for the states 1s through 7i are calculated as functions of B and C. It is shown that for a given principal quantum number n the energy eigenvalues increase (decrease) with increasing azimuthal quantum number l if the Yukawa potential is attractive (repulsive), i.e., for l>l': ${E}_{\mathrm{nl}}$\ensuremath{\ge}${E}_{\mathrm{nl}\mathcal{'}}$ if B<0, and ${E}_{\mathrm{nl}}$\ensuremath{\le}${E}_{\mathrm{nl}\mathcal{'}}$ if B>0. It leads to the crossing of the energy levels with n\ensuremath{\ge}2. For B>0 the levels with larger n and l become lower than those with smaller n and l, e.g., ${E}_{3d}$${E}_{2s}$, ${E}_{4f}$${E}_{2s}$, and ${E}_{4f}$${E}_{3p}$. For B0 and certain intervals of C the levels with larger n but smaller l lie below those with smaller n and larger l, e.g., ${E}_{4s}$${E}_{3d}$, ${E}_{5s}$${E}_{4f}$, and ${E}_{5p}$${E}_{4f}$. The values of B and C for which the lowest-energy levels cross over are estimated. Moreover, the splitting of the 2s and 2p levels (the Lamb shift) is discussed.