Analytic

How do we get what the lines are going to do if the fine structure constant was a different value?

First, derive how the energy shifts for different level transitions in a Hydrogen-like atom. A source that helped with this first part: Relativistic Quantum Mechanics by Bjorken and Drell, (c) 1964.

Hamiltonian:

(1)
\begin{align} H \Psi = \left [ \alpha \vec{p} + \beta m + V(r) \right ] \Psi = E \Psi \end{align}
(2)
\begin{align} V(r) = - \frac{Z a}{r} \end{align}
(3)
\begin{align} \vec{J}=\vec{L} + \vec{S} = \vec{r} \times \vec{p} + \frac{1}{2} \vec{\sigma} \end{align}

Commutes with H, J2, and Jz.

If $\Psi$ is a 2 component spinor

(4)
\begin{align} \Psi = \left ( \begin{array}{c} \phi \\ \chi \end{array} \right ) \end{align}

then the angular separation for $\phi$ and $\chi$ is that of Pauli 2 component theory.

The 2 component angular solutions are eigenfunctions of J2, Jz, L2, and S2 and of two types:

for:

(5)
\begin{align} j = l + \frac{1}{2} => \phi_{j,m}^{(+)} = \left [ \begin{array}{ll} \sqrt{ \frac{l+\frac{1}{2}+m}{2 l + 1}} Y_{l}^{m-\frac{1}{2}} \\ \sqrt{ \frac{l+\frac{1}{2}-m}{2 l + 1}} Y_{l}^{m+\frac{1}{2}} \end{array} \right ] \end{align}

for:

(6)
\begin{align} j = l - \frac{1}{2} => \phi_{j,m}^{(-)} = \left [ \begin{array}{ll} \sqrt{ \frac{l+\frac{1}{2}-m}{2 l + 1}} Y_{l}^{m-\frac{1}{2}} \\ \sqrt{ \frac{l+\frac{1}{2}+m}{2 l + 1}} Y_{l}^{m+\frac{1}{2}} \end{array} \right ] \end{align}

with the negative phi solution only existing for l > 0.

These functions satisfy the eigenvalue equations:

(7)
\begin{align} J^2 \phi_{jm}^{\pm} = j (j + 1) \phi_{jm} \\ \end{align}
(8)
\begin{align} \vec{L} \cdot \vec{\sigma} \phi_{jm}^{\pm} = \left ( J^2 - L^2 - \frac{3}{4} \right) \phi_{jm}^{\pm} \\ \end{align}
(9)
\begin{align} \vec{L} \cdot \vec{\sigma} \phi_{jm}^{\pm} = - (1 - \kappa ) \phi_{jm}^{\pm} \end{align}

defining:

(10)
\begin{align} \kappa=\left \{ \begin{array}{rclcl} -(l+1) &=& - (j + \frac{1}{2} ) \mbox{ , } j &=& l + \frac{1}{2} \\ + l &=& + (j + \frac{1}{2} ) \mbox{ , } j &=& l - \frac{1}{2} \end{array} \end{align}

for a given j the equations are of opposite polarity since l values differ by 1, and can be formed from each other by a scalar operator of odd parity. This operator will bea linear combination of $Y_1^m (\theta,\phi)$ since it must change the l value by 1, and is therefore proportional to r.

Dotting with pseudovector $\sigma$, to make $\vec{\sigma} \cdot \frac{\vec{r}}{r}$ you get:

(11)
\begin{align} \phi_{jm}^{(+)} = \frac{\vec{\sigma} \cdot \vec{r}}{r} \phi_{jm}^{(-)} \end{align}

The general solution to the central field problem for a given jm is:

(12)
\begin{align} \phi_{jm} = \left ( \begin{array}{ll} \frac{\mathi G_{j}^{+}}{r} \phi_{jm}^{(+)} + \frac{\mathi G_{j}^{-}}{r} \phi_{jm}^{(-)} \\ \frac{\mathi F_{j}^{+}}{r} \phi_{jm}^{(-)} + \frac{\mathi F_{j}^{-}}{r} \phi_{jm}^{(+)} \end{array} \right ) \end{align}

Where F and G are Coulomb Wave Functions.

Punchline:

(13)
\begin{align} E_n = m \left [ 1 + \left ( \frac{Z \alpha}{n - \left ( j + \frac{1}{2} \right ) + \sqrt{\left ( j + \frac{1}{2} \right )^2 - Z^2 \alpha^2} } \right )^2 \right ] ^{-\frac{1}{2}} \end{align}

Expanding in powers of $(Z \alpha)^2$:

(14)
\begin{align} E_n \approx m \left \{ 1 - \frac{1}{2} \frac{Z^2 \alpha^2}{n^2} \left [ 1 + \frac{Z^2 \alpha^2}{n} \left ( \frac{1}{j + \frac{1}{2}} - \frac{3}{4 n} \right ) \right ] + \mathcal{O}((Z \alpha)^6) \right \} \end{align}