We investigate the approximate solutions of the Dirac equation with the position-dependent mass particle in the Eckart potential field including the Coulomb tensor interaction by using the parametric Nikiforov-Uvarov method. Taking an appropriate approximation to deal with the centrifugal term, the Dirac energy states and the corresponding normalized two-spinor components of the wave function are obtained in closed form. Some special cases of our solution are investigated. Furthermore, we present the correct solutions obtained via the asymptotic iteration method which are in agreement with the parametric Nikiforov-Uvarov method results.

Bound states of spatially dependent mass Dirac equation with the Eckart potential including Coulomb tensor interaction

The Sommerfeld factor for arbitrary partial wave processes is derived in the non-relativistic limit. The s-wave and p-wave numerical results are presented for the case of Yukawa interactions. An approximate analytic expression is also found for the Sommerfeld factor of Yukawa interactions with arbitrary partial waves, which is exact in the Coulomb limit. It is demonstrated that this result is accurate to within 10% for some common scenarios. The non-s-wave Sommerfeld effect is determined to be significant, and can allow higher partial waves to dominate cross sections.

Sommerfeld factor for arbitrary partial wave processes

Analytical approximations to the solutions of the Manning-Rosen potential with centrifugal term

Approximate analytical solutions of the Dirac equation with the generalized Pöschl–Teller potential including the pseudo-centrifugal term

The bound-state solutions of the Schrodinger equation with the Eckart potential with the centrifugal term are obtained approximately. It is shown that the solutions can be expressed in terms of the generalized hypergeometric functions 2F1(a, b; c; z). The intractable normalized wavefunctions are also derived. To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other methods for short-range potential (large a). Two special cases for l = 0 and β = 0 are also studied briefly.

https://iopscience.iop.org/article/10.1088/1751-8113/40/34/010/pdf

Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential

Using solutions to a Hulth\`en-like effective potential as variational trail functions we have calculated the energy levels of the nonzero angular momentum states of the static screened Coulomb potential. Our one-parameter results for the $2p$, $3p$, $3d$, $4p$, $4d$, and $4f$ levels are in excellent agreement with earlier, more elaborate calculations. We have also calculated spontaneous emission transition probabilities between several pairs of states and find that our results compare favorably with previous calculations. We conclude that Hulth\`en-like trial functions provide better variational energies and wave functions with fewer parameters than hydrogenic or Slater-type functions for screened Coulomb and similar potentials.

C. Aldrich

Papers

Variational wave functions for a screened Coulomb potential