Fermionic particles with position-dependent mass in the presence of inversely quadratic Yukawa potential and tensor interaction

Abstract

Approximate solutions of the Dirac equation with position-dependent mass are presented for the inversely quadratic Yukawa potential and Coulomb-like tensor interaction by using the asymptotic iteration method. The energy eigenvalues and the corresponding normalized eigenfunctions are obtained in the case of position-dependent mass and arbitrary spin-orbit quantum number k state and approximation on the spin-orbit coupling term.

Appendix

To normalize the wave functions, some of the special procedures for the beta function is given in the following form [32–38]:

• (i)
  $$ B_q \left( {x+1,y} \right)=\dfrac{x}{x+y}B_q \left( {x,y} \right)-\dfrac{q^x\left( {1-q} \right)^y}{x+y}, $$
• (ii)
  $$ {\begin{array}{lll} I_q \left( {x,y} \right)&=&I_q \left( {x-1,y} \right)-\dfrac{q^{x-y}\left( {1-q}\right)^y}{x+y}\\ &=&I_q \left( {x-2,y} \right)-\dfrac{q^{x-2}\left( {1-q} \right)^y}{\left( {x-2} \right)B\left( {x-2,y} \right)}-\dfrac{q^{x-1}\left( {1-q} \right)^y}{\left( {x-1} \right)B\left( {x-1,y} \right)} \\ &=&I_q \left( {x-3,y} \right)-\dfrac{q^{x-3}\left( {1-q} \right)^y}{\left( {x-3} \right)B\left( {x-3,y} \right)}-\dfrac{q^{x-2}\left( {1-q} \right)^y}{\left( {x-2} \right)B\left( {x-2,y} \right)} \\ &&{\kern12pt}-\dfrac{q^{x-1}\left( {1-q} \right)^y}{\left( {x-1} \right)B\left( {x-1,y} \right)}\\ &=&\cdots \\ &=&I_q \left( {x-m,y} \right)-q^x\left( {1-q} \right)^y\sum\limits_{k=1}^m {\dfrac{q^{-k}}{\left( {x-k} \right)B\left( {x-k,y} \right)},} \\ m&=&1,2,..., \\ \end{array} }$$
• (iii)
  $${\begin{array}{lll} B_q{\kern-2pt}\left({x,y} \right)&=\dfrac{q^x{\kern-2pt}\left( {1-q} \right)^{y-1}}{x}\sum\limits_{k=0}^\infty {\dfrac{\left( {1-y} \right)_k }{\left( {1+x} \right)_k }} \left( {\dfrac{q}{q-1}} \right)^k \\[8pt] &=\dfrac{q^x\!\left({1\!-\!q} \right)^{y-1}}{x}{ }_2F_1 \!\left(\!{1,\!1\!-\!y;1\!+\!x;\!\dfrac{q}{q\!-\!1}} \!\right)\\[8pt] \end{array}} $$

  for $\;q\!\in\! \left({-\infty ,0} \right)\!\cup\! ({0,\frac{1}{2}})$ and

  $$ {\begin{array}{lll} B_q \left( {x,y} \right)&=&B\left( {x,y} \right)-\dfrac{q^{x-1}\left( {1-q} \right)^y}{y}\sum\limits_{k=0}^\infty {\dfrac{\left( {1-x} \right)_k }{\left( {1+y} \right)_k }} \left( {\dfrac{q-1}{q}} \right)^k \\ &=&B\left( {x,y} \right)-\dfrac{q^{x-1}\left( {1-q} \right)^y}{y}{ }_2F_1 \left( {1,1-x;1+y;\dfrac{q-1}{q}} \right), \\ \end{array} } $$

• (iv) $\left( a \right)_{i+j} =\left( a \right)_i \left( {a+i} \right)_j .$