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Please try to include the following

• main points understood, and expand them - what is your understanding of what the points were.
  • expand these points by including many of the details the class discussed.
• main points which are not clear. - describe what you have understood and what the remain questions surrounding the point(s).
  • Other classmates can step in and clarify the points, and expand them.
• How the main points fit with the big picture of QM. Or what is not clear about how today's points fit in in a big picture.
• wonderful tricks which were used in the lecture.
  

Fine structure is due to two mechanisms: relativistic correction and spin-orbit coupling. In other words, a very small perturbation (correction) to the Bohr energies.

The equation of which is: <math>H_{fs}=\alpha\vec{S}\vec{L}</math>

By considering the relativistic version of momentum: <math>p=\frac{mv} {\sqrt{1-(\frac v {c})^2}}</math>. We can derive the relativistic equation for kinetic energy: <math>T=\sqrt{m^2c^4+p^2c^2}-mc^2</math>.

With <math>p«mc</math> (the non-relativistic limit) we get <math>T=\frac {p^2} {2m}-\frac{p^4} {8m^3c^2}</math>

This gives us the lowest-order relativistic correction <math>H'_r=H-H^0</math> with <math>H'_r=\frac{-p^4} {8m^3c^2}</math>

With this we can then move to find <math>E^1_r</math> which is equal to: <math><H'_r></math> which is equal to: <math>-\frac{1} {8m^3c^2} <p^2\psi|p^2\psi></math>

If <math>p^2\psi=2m(E-V)\psi</math> then the prior equation gives us an <math><\psi|(E-V)^2|\psi></math> to deal with. Following Hermitian Operator rules we get:

<math><\psi|E^2|\psi>=E^2</math>

<math><\psi|EV|\psi>=E<V> \appr E<\frac{1} r> \appr \frac{E} {n^2a}</math>

<math><\psi|V^2|\psi> \appr <\frac{1} {r^2}> \appr \frac{1} {(l+\frac{1} 2)n^3a^2}</math>

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