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6.
$$ \begin{align} H = \sqrt{ \mathbf{p}^2 c^2 + \mu^2 c^4} + V(\mathbf{r} ) & \Rightarrow \left ( H -V(\mathbf{r} ) \right ) ^2 = \mathbf{p}^2 c^2 + \mu^2 c^4 \\ & \Rightarrow \left ( H -V( \mathbf{r} ) \right ) ^2 | \psi ( \mathbf{r} ) \rangle = \left ( \mathbf{p}^2 c^2 + \mu^2 c^4 \right ) | \psi (\mathbf{r} ) \rangle \\ \end{align}$$

The potential is rotationally invariant. The Hamiltonian operator is only the function of $$\hat{\mathbf{R}}^2 $$ and $$\hat{\mathbf{P}}^2 $$.

So the Hamiltonian and Angular momentum operators mutually commute, which means that $$| l, m \rangle $$ are eigenstates. $$ \Rightarrow \psi _E (\mathbf{r}) = R_{El} Y_l^m $$

$$ \begin{align} \left ( E - V(r) \right ) ^2 R_{El} Y_l^m & = -\hbar^2 c^2\nabla ^2 R_{El} Y_l^m + \mu ^2 c^4 R_{El} Y_l ^m = -\hbar^2 c^2 \left [ { 1\over r^2} {\partial \over { \partial r}} \left ( r^2 {\partial \over {\partial r}} \right ) - {1\over {\hbar^2 r^2}} \hat{\mathbf{L}}^2 \right ] R_{El} Y_l^m + \mu^2 c^4 R_{El} Y_l^m \\ & = -\hbar^2 c^2 \left [ { 1\over r^2} {\partial \over { \partial r}} \left ( r^2 {\partial \over {\partial r}} \right ) - {{l \left ( l + 1 \right ) }\over r^2}\right ] R_{El} Y_l^m + \mu^2 c^4 R_{El} Y_l^m \end{align} $$

Dividing each side with $$Y_l^m$$ we get,

$$ \begin{align} \left ( E - V(r) \right ) ^2 R_{El} & = -\hbar^2 c^2 \left [ { 1\over r^2} {d \over { dr}} \left ( r^2 {d \over {dr}} \right ) - {{l \left ( l + 1 \right ) }\over r^2}\right ] R_{El} + \mu^2 c^4 R_{El} \end{align} $$

Let $$ U_{El} = r R_{El} $$.

$$\left ( E - V(r) \right ) ^2 U_{El} = \left ( E + {e^2 \over {4 \pi \epsilon_0 r}} \right ) ^2 U_{El} = \left [ -\hbar^2 c^2{d^2\over {dr^2}} + {{l \left ( l + 1 \right ) \hbar^2 c^2} \over {r^2}} \right ] U_{El} + \mu ^2 c^4 U_{El} $$

$$ \left [ {d^2 \over {dr^2}} - {{l\left ( l + 1 \right ) } \over r^2} + {1 \over {\hbar^2 c^2}} \left ( E + {e^2 \over {4 \pi \epsilon_0 r}} \right ) ^2 - {{\mu^2 c^2} \over \hbar^2} \right ] U_{El} = 0$$

$$ \left [ {d^2 \over {dr^2}} - {{l\left ( l + 1 \right ) } \over r^2} + \left ( {e^2 \over {4 \pi \epsilon_0 \hbar c r}} \right ) ^2 + {1 \over {\hbar^2 c^2}} \left ( E^2 + {e^2 E \over {2 \pi \epsilon_0 r}} - \mu^2 c^4 \right ) \right ] U_{El} = 0 $$

$$ \left [ {d^2 \over {dr^2}} - \left \{ l\left ( l + 1 \right ) - \left ( {e^2 \over {4 \pi \epsilon_0 \hbar c }} \right ) ^2 \right \} {1 \over r^2} + {1 \over {\hbar^2 c^2}} \left ( E^2 +  {e^2 E \over {2 \pi \epsilon_0 r}} - \mu^2 c^4 \right ) \right ] U_{El} = 0 $$

Take $$ \kappa ^2 = {1 \over {\hbar^2 c^2}} \left ( \mu^2 c^4 - E^2 \right )$$ and $$ l\left ( l + 1 \right )  - \left ( {e^2 \over {4 \pi \epsilon_0 \hbar c }} \right ) ^2 = l' \left ( l' + 1 \right ) $$.

$$ \left [ {d^2 \over {dr^2}} - {{l' \left ( l' + 1 \right ) } \over r^2} - \kappa ^2 + {e^2 E \over {2 \pi \epsilon_0 \hbar^2 c^2 r}} \right ] U_{El} = 0 $$

As $$r \to \infty $$ the equation becomes $$ {d^2 \over {dr^2}} U_{El} - \kappa^2 U_{El} = 0 $$.

The solution to this equation is $$ Ae^{-\kappa r} $$. Take $$ \rho = \kappa r$$ and $$U_{El} = e^{-\rho}v_{El}$$.

$$ \left [ \kappa^2 {d^2 \over {d\rho^2}} - \kappa ^2 {{l' \left ( l' + 1 \right ) } \over \rho^2} - \kappa ^2 + \kappa {e^2 E \over {2 \pi \epsilon_0 \hbar^2 c^2 \rho}} \right ] v_{El} e^{-\rho} = 0 $$

$$ \left [ {d^2 \over {d\rho^2}} - {{l' \left ( l' + 1 \right ) } \over \rho^2} - 1 + {e^2 E \over {2 \pi \epsilon_0 \hbar^2 c^2 \kappa \rho}} \right ] v_{El} e^{-\rho} = 0 $$

$$ {d^2 v_{El} \over {d\rho^2}} e^{-\rho} - 2 {d v_{El} \over {d\rho}} e^{-\rho} + v_{El} e^{-\rho} + \left [ - {{l' \left ( l' + 1 \right ) } \over \rho^2} - 1 + {e^2 E \over {2 \pi \epsilon_0 \hbar^2 c^2 \kappa \rho}} \right ] v_{El} e^{-\rho} = 0 $$

$$ {d^2 v_{El} \over {d\rho^2}} - 2 {d v_{El} \over {d\rho}} + v_{El} + \left [ - {{l' \left ( l' + 1 \right ) } \over \rho^2} - 1 + {e^2 E \over {2 \pi \epsilon_0 \hbar^2 c^2 \kappa \rho}} \right ] v_{El} = 0 $$

$$ {d^2 v_{El} \over {d\rho^2}} - 2 {d v_{El} \over {d\rho}} - \left [ {{l' \left ( l' + 1 \right ) } \over \rho^2} - {e^2 E \over {2 \pi \epsilon_0 \hbar^2 c^2 \kappa \rho}} \right ] v_{El} = 0 $$

Take $$\lambda = {E \over {2 \pi \epsilon_0 \hbar^2 c^2 \kappa }}$$.

$$ {d^2 v_{El} \over {d\rho^2}} - 2 {d v_{El} \over {d\rho}} - \left [ {{l' \left ( l' + 1 \right ) } \over \rho^2} - {e^2 \lambda \over \rho} \right ] v_{El} = 0 $$

This equation is same as (13.1.8). $$ \Rightarrow e^2 \lambda = 2 \left ( k + l' + 1 \right ) $$.

$$\left ( {e^2 \over {2 \pi \epsilon_0 \hbar^2 c^2}} \right ) ^2 E^2 = 4 \left ( k + l' + 1 \right ) ^2 \kappa ^2$$

$$\left ( {e^2 \over {4 \pi \epsilon_0 \hbar c}} \right ) ^2 E^2 = \left ( k + l' + 1 \right ) ^2 \left ( \mu^2 c^4 -E^2 \right )$$

Let $${e^2 \over {4 \pi \epsilon_0 \hbar c}} = \alpha $$.


 * $$\therefore E_k^2 = {\mu^2 c^4 \left ( k + l' + 1 \right ) ^2 \over {\alpha ^2 + \left ( k + l' + 1 \right ) ^2 }} $$ where $$k\in\mathbb{N}$$



l\left ( l + 1 \right ) - \left ( {e^2 \over {4 \pi \epsilon_0 \hbar c }} \right ) ^2 = l \left ( l + 1 \right ) - \alpha ^2 = l' \left ( l' + 1 \right ) \Rightarrow l'^2 + l' +\alpha ^2 - l \left ( l + 1 \right ) = 0 \Rightarrow l' = -{1\over 2} + \sqrt{l \left ( l + 1 \right ) - \alpha^2 + {1\over 4}}$$


 * $$\alpha \approx {1 \over 137}$$, so $$l'$$ always exists and is real.