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\begin{document}

The fine structure of Hydrogen is derivably by perturbation theory.

\( H_{0}=\frac{p^{2}}{2\mu }-\frac{Ze^{2}}{r} \) Where \( \mu =\frac{m_{e}m_{p}}{m_{e}+m_{p}} \) The unperturbed solution is \( <r^{i},\sigma |nlm_{l}m_{s}>=R_{nl}(r)Y_{lm_{l}}\delta _{\sigma m_{s}} \). \( R_{nl}(r)=\frac{1}{a^{\frac{3}{2}}}f_{nl}(\frac{r}{a}) \). These staisfy: \( \int Y_{lm}^{*}Y_{l'm'}=\delta _{ll'}\delta _{mm'} \) and \( \int r^{2}drR_{nl}R_{n'l'}=\delta _{nn'} \).
The energy is \( E_{n}^{(0)}=\frac{-Z^{2}}{n^{2}}R_{y} \) and \( R_{y}=\frac{\mu e^{4}}{2\hbar ^{2}} \)

Relativistic corrections:

Start with dirac equation and expand in powers of \( \frac{v}{c}\sim \alpha  \). 

\begin{itemize}

\item First correction due to relativistic mass. \( \sqrt{(\mu c^{2})^{2}+(pc)^{2}}-\mu c^{2}=\frac{p^{2}}{2\mu }-\frac{1}{2\mu c^{2}}(\frac{p^{2}}{2\mu })^{2}+higherorder. \) Correction is of order:
\( Z^{2}\alpha ^{2} \).

\item Spin orbit interaction. \( H_{LS}=\frac{1}{2}(\frac{\hbar }{\mu c})^{2}(\frac{1}{r}\frac{\partial V}{\partial r})\frac{L^{i}S^{i}}{\hbar ^{2}} \) \( =2Z(\frac{\hbar e}{2\mu c})^{2}\frac{1}{r^{3}}\frac{L^{i}S^{i}}{\hbar ^{2}} \) This is also order \( Z^{2}\alpha ^{2} \). Doesn't contribute
for S-waves.

\item Darwin term-has to do with correction due to only local interactions.
\( H_{D}=\frac{\hbar ^{2}}{8\mu ^{2}c^{2}}\nabla ^{i}\nabla ^{i}V(r)=Z\frac{\hbar ^{2}e^{2}}{8\mu ^{2}c^{2}}4\pi \delta (r) \). This is oreder \( Z^{2}\alpha ^{2} \). Only contributes to S-waves.

\end{itemize}

These are the fine structure.

In uncoupled basis, only spin-orbit term is nondiagonal. In coupled
basis, everything is diagonal.

Then \( |nljm_{j}>=\sum _{m_{l}m_{j}}|nlsm_{l}m_{s}><l\frac{1}{2}m_{l}m_{s}|jm_{j}> \) which gives \( \Psi _{nljm_{m}}=R_{nl}Y_{jlm_{j}} \). Then for the relativistic mass, \( \triangle E=\frac{-1}{2\mu c^{2}}<jm_{J}|(\frac{p^{2}}{2\mu })^{2}|jm_{j}> \) \( =\frac{-1}{2\mu c^{2}}<jm_{J}|(E_{n}+\frac{Ze^{2}}{r})^{2}|jm_{j}> \)\( =\frac{-1}{2\mu c^{2}}[E_{n}^{2}+2E_{n}Ze^{2}<\frac{1}{r}>+Z^{2}e^{4}<\frac{1}{r^{2}}>] \)

\( =\frac{-1}{2\mu c^{2}}[E_{n}^{2}+2E_{n}Ze^{2}\frac{1}{n^{2}a_{0}}+Z^{2}e^{4}\frac{1}{(2l+1)n^{3}a_{0}^{2}}] \) \( =-|E_{n}|\frac{Z^{2}\alpha ^{2}}{n^{2}}(\frac{2n}{2l+1}-\frac{3}{4}) \)

For the spin-orbital interaction: \( \triangle E=<H_{LS}>=\frac{Z}{4}(\frac{\hbar e}{\mu c})^{2}(j(j+1)-l(l+1)-\frac{3}{4})\frac{2}{l(l+1)(2l+1)n^{3}a^{3}} \)

\( |E_{n}|\frac{Z^{2}\alpha ^{2}}{n^{2}}\begin{array}{cc}
\frac{-n}{(2j+1)(j+1)} & j=l-\frac{1}{2}\\
\frac{n}{j(2j+1)} & j=l+\frac{1}{2}
\end{array}
 \)

For the Darwinian interaction: \( \triangle E_{D}=|E_{n}|\frac{Z^{2}\alpha ^{2}}{4n}\delta _{l0} \)

The sum of the terms is then: \( \triangle E_{tot}=|E_{n}|[1+\frac{Z^{2}\alpha ^{2}}{n^{2}}(\frac{n}{j+\frac{1}{2}}-\frac{3}{4})] \). There is only j angular dependence.
For n=2, you get, \( 2p_{\frac{3}{2}} \) splitting from \( 2s_{\frac{1}{2}},2p_{\frac{1}{2}} \) by about \( 5*10^{-5}eV \).

hyperfine structure and lamb shift add in.

\end{document}
