Retarded Hydrogen Atom
The non-relativistic Hydrogen atom has the following spectrum,
High resolution spectrometers observed a fine splitting of the above degeneracy. What follows below is a more detailed analysis of the Hydrogen atom Hamiltonian to explain the fine structure.
Baby Dirac equation
How do we reconcile the interaction of electron in a hydrogen atom with the magnetic field? Here's how people have tried formulating.
For a loop carrying a current I, we write the magnetic moment $\bar{\mu} = \frac{I \bar{A}}{C}$. So, an electron with orbital angular momentum $\bar{L}$ has $\bar{\mu} = \frac{q}{2mC}\bar{L}$.
Being reasonably intuitive, people have tried writing $\bar{\mu} = g \frac{e}{2m_eC} \bar{S}$ for the magnetic moment due to the quantum spin of electron. Clearly we were ambitious about the nature of magnetic moment in quantum regime in writing it almost same as classical one. But we have also parametrized all our ignorance in the factor $g$. Experiments show that $g \approx 2$, a simple 2 for the electron! So quantum spin is not that strange from classical after all! But nevertheless this is the first glance at why the quantum spin is not about electron spinning around it's axis, although telling us its probably okay to think on those lines (with a whispering $g$ accompanying every time).
So, Pauli in 1927 went on to formulate this whole business. He noticed something funny, that \begin{align} (\bar{\sigma} .\bar{a}) (\bar{\sigma} .\bar{b}) = (\bar{a}.\bar{b} +i\bar{\sigma}. (\bar{a}\times\bar{b})) \tag{1} \end{align} gives $(\bar{\sigma} .\bar{p})^2 = \bar{p}^2$. With prior learning from Stern-Gerlach eperiment regarding the two spin states of electron, he started out with writing the Hamiltonian eigenstate as a column vector, a 2-spinor; $\psi = \begin{bmatrix} \chi \\ \eta \end{bmatrix}$.
Rewriting the hamiltonian this way replacing $\bar{p}^2$ with $(\bar{\sigma}.\bar{p})^2$ and in present of the E.M fields, $\bar{p} \rightarrow \bar{\pi} = \bar{p} - \frac{q}{C}\bar{A}$, we have \begin{align} H_{Pauli} &= {\frac {1}{2m}}\left[{\bar {\sigma }}\cdot (\bar {p} -q\bar {A} )\right]^{2}+q\phi \tag{2} \\ &= \frac{1}{2m}\left(\bar {p} -q\bar {A} \right)^{2}-2\times\frac{q\hbar {\bar {\sigma }}\cdot \bar {B}}{2m}+q\phi \\ &= {\frac {1}{2m}}\left[|\bar {p} |^{2}-q(\bar {L} +2\bar {S} )\cdot \bar {B} \right]+q\phi \tag{3} \end{align} Where we have assumed magnetic field is weak in writing the third line. The g-factor is naturally seen to be 2 in the Pauli equation as the coefficient of $\bar{S}$.
The Dirac equation
The Electromagnetic interaction is reconciled due to Pauli, and even the Lande's g-factor for the electron is explained! In 1928 Dirac figured a way to blend the relativistic artifacts correctly with the Pauli's equation. This is a nice point to discuss the issues in relativistic formulation that physicists faced which ultimately lead to QFT. I divert all such subtleties to a blog titled The beginnings of QFT, but for now I will assume Dirac equation is a nice way to reconcile the relativistic mechanics in the quantum world. To escape the infinite orders of terms when one expands the relativistic energy, Dirac wrote $p^2C^2 + m^2C^4$ as a square of some linear function of $p$. Clearly there are no mixing terms, so Dirac went crazy. He wrote, $$p^2C^2 + m^2C^4 = (C\bar{\alpha}.\bar{p} + \beta mC^2)$$ Doing the obvious sanity checks leaves us with $(\alpha_i)^2 = \beta^2 = 1; \alpha_i\beta + \beta\alpha_i = 0 = \alpha_i\alpha_j + \alpha_j\alpha_i$. With nice little discussion on when will such properties be satisfied, one ends up with the following representation, \begin{align} \bar{\alpha} = \begin{bmatrix} 0 & \bar{\sigma} \\ \bar{\sigma} & 0 \end{bmatrix}; \beta = \begin{bmatrix} \mathbb{1} & 0 \\ 0 & -\mathbb{1}\end{bmatrix} \end{align}
Foldy–Wouthuysen transformation
I understand only the following as of now:Writing $\psi = \Omega^{-1}\chi$ with $\Omega = 1 + \frac{(\bar{\sigma}.\bar{\pi})^2}{8m^2C^2}$ gives the right Fine structure Hamiltonian.
The Fine structure of Hydrogen atom
Cut to post Barton Zwiebach lecture binge. Here's the Hydrogen Atom Fine structure summarized.
My complete handwritten notes of the lectures by Barton Zwebach, MIT OCW, 8.06 on the Fine structure of Hydrogen atom can be found here. A not-so-complete notes on Foldy–Wouthuysen transformation oof lectures from P40X, NISER can be found here.
Zeeman Structure
In explaining the Fine strcture of Hydrogen atom, we used the Dirac equation and added an electromagnetic interaction term, eventually assuming only the internal coloumbic electric field to be present i.e. no other external electric or magnetic field. In this section we will introduce an external magnetic field.
The strategy is as follows - Start with the good old not so fine Hamiltonian and