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\begin{document}

The anomalous Zeeman effect is splitting into an even number of levels
(not predicted by angular momentum alone) for Z odd atoms. This is
evidence for spin. \( H=H_{0}-g\mu _{B}S_{i}B_{i} \)is the hamiltonian of the effect. \( g=2.0001 \). 

If you turn off the magnetic field, there is still an effect:\( H=H_{0}-f(r)g\mu _{B}^{2}L_{i}S_{i} \) because
of the magnetic field generated by the moving electrons.

\( \triangle E=<jlm_{j}|\mu _{B}(J_{z}+S_{z})B_{z}|jlm_{j}>=\mu _{B}B_{z}m_{j}+\mu _{B}B_{z}<jlm_{j}|S_{z}|jlm_{j}> \)

The wigner eckhart theorem is \( <jlm_{j}|V^{i}|jlm_{j}>=\frac{1}{2j+1}<||V||><J1m_{j}q|jm_{j}> \)

so \( <jlm_{j}|V_{i}|jlm_{j}> \)= \( \frac{<||V||>}{<||J||>}<jlm_{j}|J_{i}|jlm_{j}> \) so all vector operators are proportional to \( <J_{i}> \)

Want to find \( <||V||>/<||J||> \). Consider \( <jlm_{j}|V_{i}J^{i}|jlm_{j}>\sum _{m'}<jlm|V_{i}|j'l'm'><j'l'm'|J^{i}|jlm> \) \( =\frac{<||V||>}{<||J||>}<jlm|J_{i}J^{i}|jlm>= \)\( =\frac{<||V||>}{<||J||>}(j(j+1)) \). This means: \( <jlm_{j}|V_{i}|jlm_{j}>=<jlm_{j}|J_{i}|jlm_{j}>\frac{<jlm_{j}|V_{i}J^{i}|jlm_{j}>}{j(j+1)} \)y.

in our paricular case: \( \mu ^{i}J_{i}=\mu _{B}(J_{i}J^{i}+\frac{1}{2}(S_{i}S^{i}+J_{i}J^{i}-L_{i}L^{i})) \). so \( <\mu _{z}B_{z}>=\frac{m_{j}\mu _{B}B_{z}}{j(j+1)}[j(j+1)+\frac{3}{4}+\frac{1}{2}(j(j+1)+\frac{3}{4}-l(l+1))]=g_{jls}m_{j}\mu _{B}B_{z} \) for \( g_{jls}=[\frac{\frac{3}{2}j(j+1)+\frac{3}{8}-\frac{1}{2}(l(l+1)}{j(j+1)}] \)= lande g-factor.

\end{document}
