Approximate bound and scattering solutions of Dirac equation for the modified deformed Hylleraas potential with a Yukawa-type tensor interaction

Abstract

Analytical bound and scattering state solutions of Dirac equation are investigated for the modified deformed Hylleraas potential with a Yukawa-type tensor interaction. The energy equation, phase shifts and normalization constants of the pseudospin and spin symmetry limits are represented. Since the modified deformed Hylleraas potential reduces to the Pöschl–Teller, Hulthén and deformed Hylleraas potential, we have obtained energy equation and scattering properties of the Dirac equation for these potentials within a Yukawa-type tensor interaction. We have also reported some numerical results to show the effect of tensor interaction.

Appendix

The NU method solves many linear second order differential equations by reducing them to a generalized equation of hypergeometric type. Here, instead of the original formulation, we use the parametric version which enables us to solve a second-order differential equation of the form [27, 28, 29]

$$ \left\{ {\frac{{d^{2} }}{{ds^{2} }} + \frac{{\alpha_{1} - \alpha_{2} s}}{{s(1 - \alpha_{3} s)}}\frac{d}{ds} + \frac{1}{{[s(1 - \alpha_{3} s)]^{2} }}[ - \xi_{1} s^{2} + \xi_{2} s - \xi_{3} ]} \right\}\psi = 0 $$

(77)

According to the NU method, the eigenfunctions is

$$ \psi (s) = s^{{\alpha_{12} }} (1 - \alpha_{3} s)^{{ - \alpha_{12} - \frac{{\alpha_{13} }}{{\alpha_{3} }}}} P_{n}^{{\left( {\alpha_{10} - 1,\frac{{\alpha_{11} }}{{\alpha_{3} }} - \alpha_{10} - 1} \right)}} (1 - 2\alpha_{3} s) $$

(78)

and the energy of the system satisfies

$$ \alpha_{2} n - (2n + 1)\alpha_{5} + (2n + 1)(\sqrt {\alpha_{9} } + \alpha_{3} \sqrt {\alpha_{8} } ) + n(n - 1)\alpha_{3} + \alpha_{7} + 2\alpha_{3} \alpha_{8} + 2\sqrt {\alpha_{8} \alpha_{9} } = 0, $$

(79)

where

$$ \begin{aligned} \alpha_{4} = \frac{1}{2}(1 - \alpha_{1} ),\quad \alpha_{5} = \frac{1}{2}(\alpha_{2} - 2\alpha_{3} ),\quad \alpha_{6} = \alpha_{5}^{2} + \xi_{1} ,\quad \alpha_{7} = 2\alpha_{4} \alpha_{5} - \xi_{2} ,\quad \alpha_{8} = \alpha_{4}^{2} + \xi_{3} \hfill \\ \alpha_{9} = \alpha_{3} \alpha_{7} + \alpha_{3}^{2} \alpha_{8} + \alpha_{6} ,\quad \alpha_{10} = \alpha_{1} + 2\alpha_{4} + 2\sqrt {\alpha_{8} } \hfill \\ \alpha_{11} = \alpha_{2} - 2\alpha_{5} + 2(\sqrt {\alpha_{9} } + \alpha_{3} \sqrt {\alpha_{8} } ),\quad \alpha_{12} = \alpha_{4} + \sqrt {\alpha_{8} } \hfill \\ \alpha_{13} = \alpha_{5} - (\sqrt {\alpha_{9} } + \alpha_{3} \sqrt {\alpha_{8} } ) \hfill \\ P_{n}^{(\alpha ,\beta )} (x) = \frac{\varGamma (\alpha + n + 1)}{n!\varGamma (\alpha + \beta + n + 1)}\sum\limits_{m = 0}^{n} {\left( \begin{aligned} n \hfill \\ m \hfill \\ \end{aligned} \right)} \frac{\varGamma (\alpha + \beta + n + m + 1)}{\varGamma (\alpha + m + 1)}\left( {\frac{x - 1}{2}} \right)^{m} \hfill \\ \end{aligned} $$

(80)

And $P_{n}^{(\alpha ,\beta )}$ is Jacobi polynomial.