Bound states and energy eigenvalues of a radial screened Coulomb potential

Abstract

We analyze bound states and other properties of solutions of a radial Schrödinger equation with a new screened Coulomb potential. In particular, we employ hypervirial relations to obtain eigen-energies for a Hydrogen atom with this potential. Additionally, we appeal to a sharp estimate for a modified Bessel function to estimate the ground state energy of such a system. Finally, when the angular quantum number ℓ ≠ 0, we obtain evidence for a critical screening parameter, above which bound states cease to exist.

Revisiting the energy spectrum of the radial screened Coulomb potential

“…The computational details and essential input parameters were introduced in our previous work [23] and they are omitted for simplicity. It can be seen from the comparison that the results obtained by Stachura and Hancock [22] are accurate within all their reported digits.…”

“…which is a natural result of equation (22). The agreement improves at larger values of C, while it slightly deteriorates for higher orbital angular momentum l, which can be understood from equations (20) and (21) where we only employ the simplest Padé[0,1] approximant in approximating the Taylor series.…”

“…This large-parameter asymptotic law is of great interest due to its close connection with the fundamental problem of how many bound states does a potential support? This question was originally proposed by Stachura and Hancock [22], while at present no solid conclusion can be drawn yet. If, however, the above inverse linear law holds strictly for all bound states of the RSCP, it gives a natural consequence that the RSCP supports an infinite number of bound states for any finite values of screening parameter C. This property is counterintuitively different from the widely used Yukawa screened-type Coulomb potential.…”

“…With a further examination of the work by Stachura and Hancock [22], it was found that the radial equation solved by those authors in the s-wave symmetry (see equation (4.37) in [22]) was given by…”

“…The energy spectrum of the RSCP has been investigated by Stachura and Hancock [22] using both analytical and numerical approaches, which include the hypervirial theorem, the variational method, and the extended Nikiforov-Uvarov method. In our previous work [23], we calculated the eigenenergies of this potential by employing the generalized pseudospectral (GPS) method [24,25] and found significant differences in the comparison with Stachura and Hancock [22]ʼs results. Such a large difference is realized to be related to the use of different Hamiltonian operators in formulating the radial Schrödinger equation.…”