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 easy.

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

The $S_z$ is harder. 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 $\left\vert\psi_{n\ell m_\ell sm_s}\right>$.

$$\bgroup\color{black} E^{(1)}_n = \left<\psi_{nj\ell m_j}\left... ...\over{2mc}}(J_z+S_z)\right\vert \psi_{nj\ell m_j}\right>\egroup$$

$$\bgroup\color{black} = {eB\over{2mc}}\left( m_j \hbar+\left<\... ...eft\vert S_z \right\vert \psi_{nj\ell m_j}\right>\right)\egroup$$

from Ch.15

$$\bgroup\color{black}\psi_{n(\ell+{1\over 2})\ell (m+{1\over 2... ... = \alpha Y_{\ell m}\chi_+ + \beta Y_{\ell (m+1)}\chi_- \egroup$$

$$\bgroup\color{black}\psi_{n(\ell-{1\over 2})\ell (m+{1\over 2... ... = \beta Y_{\ell m}\chi_+ - \alpha Y_{\ell (m+1)}\chi_- \egroup$$

$$\bgroup\color{black}\alpha=\sqrt{\ell +m+1\over {2\ell + 1}} \qquad \beta=\sqrt{\ell -m\over {2\ell + 1}} \egroup$$

$j=\ell+{1\over 2}$:

$$\bgroup\color{black}\left<\psi_{nj\ell m_j}\left\vert S_z\rig... ...chi_+ + \beta Y_{\ell (m_j+{1\over 2})} \chi_- \right> \egroup$$

$$\bgroup\color{black} = {1\over 2}\hbar\left(\alpha^2 -\beta^2\right)_{m=m_j -{1\over 2}}\egroup$$

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

$$\bgroup\color{black}\left<\psi_{nj\ell m_j}\left\vert S_z\rig... ...\pm {\hbar\over 2} {\ell + m+1-\ell + m\over{2\ell +1}} \egroup$$

$$\bgroup\color{black} = \pm {\hbar\over 2} {2(m_j -{1\over 2})+1\over {2\ell + 1}} = \pm {m_j \hbar\over{2\ell + 1}} \egroup$$

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.

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

To this we add the anomalous Zeeman effect

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