The Anomalous Zeeman Effect

We compute the energy change due to a weak magnetic field using first order Perturbation Theory.

$$\bgroup\color{black}\left<\psi_{n\ell jm_j}\left\vert{eB\over{2mc}}(L_z+2S_z)\right\vert\psi_{n\ell jm_j}\right>\egroup$$

$$\bgroup\color{black}(L_z+2S_z)=J_z+S_z\egroup$$

The $J_z$ part is easy since we are in eigenstates of that operator.

$$\bgroup\color{black}\left<\psi_{n\ell jm_j}\left\vert{eB\over... ...\vert\psi_{n\ell jm_j}\right> = {eB\over{2mc}}\hbar m_j\egroup$$

The $S_z$ is harder since we are not in eigenstates of that one. We need $\left<\psi_{n\ell jm_j}\left\vert{eB\over{2mc}}S_z\right\vert\psi_{n\ell jm_j}\right>$, but we don't know how $S_z$ acts on these. So, we must write $\left\vert\psi_{njm_j\ell s}\right>$ in terms of $\left\vert\psi_{n\ell m_\ell sm_s}\right>$.

$$\begin{eqnarray*} E^{(1)}_n &=& \left<\psi_{nj\ell m_j}\left\vert {eB\over{2mc}... ...l m_j}\left\vert S_z \right\vert \psi_{nj\ell m_j}\right>\right) \end{eqnarray*}$$

We already know how to write in terms of these states of definite $m_\ell$ and $m_s$.

$$\begin{eqnarray*} \psi_{n(\ell+{1\over 2})\ell (m+{1\over 2})} &=& \alpha Y_{\e... ...+1\over {2\ell + 1}} \\ \beta&=&\sqrt{\ell -m\over {2\ell + 1}} \end{eqnarray*}$$

Let's do the $j=\ell+{1\over 2}$ state first.

$$\begin{eqnarray*} \left<\psi_{nj\ell m_j}\left\vert S_z\right\vert \psi_{nj\ell ... ...\over 2}\hbar\left(\alpha^2 -\beta^2\right)_{m=m_j -{1\over 2}} \end{eqnarray*}$$

For $j=\ell-{1\over 2}$,

$$\bgroup\color{black}\left<\psi_{nj\ell m_j}\left\vert S_z\rig... ...}\hbar\left(\beta^2 -\alpha^2\right)_{m=m_j -{1\over 2}}\egroup$$

We can combine the two formulas for $j=\ell\pm{1\over 2}$.

$$\begin{eqnarray*} \left<\psi_{nj\ell m_j}\left\vert S_z\right\vert \psi_{nj\ell ... ...over 2})+1\over {2\ell + 1}} = \pm {m_j \hbar\over{2\ell + 1}} \end{eqnarray*}$$

So adding this to the (easier) part above, we have

$$\bgroup\color{black} E^{(1)}_n ={eB\over {2mc}}\left(m_j\hbar... ...r B\over {2mc}}m_j \left(1\pm {1\over{2\ell +1}}\right) \egroup$$

for $j=\ell\pm{1\over 2}$.

In summary then, we rewrite the fine structure shift.

$$\bgroup\color{black}\Delta E=-{1\over 2}mc^2\left(Z\alpha\rig... ...n^3}}\left[ {1\over{j+{1\over 2}}} -{3\over{4n}}\right].\egroup$$

To this we add the anomalous Zeeman effect

$$\bgroup\color{black}\Delta E = {e\hbar B\over {2mc}}m_j \left(1\pm {1\over{2\ell +1}}\right).\egroup$$