Iterative solutions to the Dirac equation

We consider a single particle which is bound by a central potential and obeys the Dirac equation in $d$ dimensions. We first apply the asymptotic iteration method to recover the known exact solutions for the pure Coulomb case. For a screened Coulomb potential and for a Coulomb plus linear potential with linear scalar confinement, the method is used to obtain accurate approximate solutions for both eigenvalues and wave functions.

Figure 1

Dirac radial functions $G\left(r\right)$ and $F\left(r\right),r>0$, in dimensionless units for $V\left(r\right)=-1∕2r+0.1r$, $U\left(r\right)=0.2r$, $k=-2$, $j=3∕2$, $n=2$. This state can be described by the spectroscopic convention as $3p_{3∕2}$.

Figure 2

Dirac spinor orbit $\mathbf{(}F\left(r\right),G\left(r\right)\mathbf{)},r>0$, in dimensionless units for the same example as in Fig. 1.