Editing Zeeman effect (section)

==Weak field (Zeeman effect)==
If the [[spin–orbit interaction]] dominates over the effect of the external magnetic field, <math>\vec L</math> and <math>\vec S</math> are not separately conserved, only the total angular momentum <math>\vec J = \vec L + \vec S</math> is. The spin and orbital angular momentum vectors can be thought of as [[precession|precessing]] about the (fixed) total angular momentum vector <math>\vec J</math>. The (time-)"averaged" spin vector is then the projection of the spin onto the direction of <math>\vec J</math>:
<math display="block">
 \vec S_\text{avg} = \frac{(\vec S \cdot \vec J)}{J^2} \vec J,
</math>
and for the (time-)"averaged" orbital vector:
<math display="block">
 \vec L_\text{avg} = \frac{(\vec L \cdot \vec J)}{J^2} \vec J.
</math>

Thus
<math display="block">
 \langle V_\text{M} \rangle = \frac{\mu_\text{B}}{\hbar} \vec J
  \left(g_L\frac{\vec L \cdot \vec J}{J^2} + g_S\frac{\vec S \cdot \vec J}{J^2}\right) \cdot \vec B.
</math>
Using <math>\vec L = \vec J - \vec S</math> and squaring both sides, we get
<math display="block">
 \vec S \cdot \vec J = \frac{1}{2} (J^2 + S^2 - L^2) = \frac{\hbar^2}{2} [j(j + 1) - l(l + 1) + s(s + 1)],
</math>
and using <math>\vec S = \vec J - \vec L</math> and squaring both sides, we get
<math display="block">
 \vec L \cdot \vec J = \frac{1}{2} (J^2 - S^2 + L^2) = \frac{\hbar^2}{2} [j(j + 1) + l(l + 1) - s(s + 1)].
</math>

Combining everything and taking <math>J_z = \hbar m_j</math>, we obtain the magnetic potential energy of the atom in the applied external magnetic field:
<math display="block">\begin{align}
 V_\text{M} 
  &= \mu_\text{B} B m_j \left[g_L \frac{j(j + 1) + l(l + 1) - s(s + 1)}{2j(j + 1)} +
                              g_S \frac{j(j + 1) - l(l + 1) + s(s + 1)}{2j(j + 1)}\right] \\
  &= \mu_\text{B} B m_j \left[1 + (g_S - 1) \frac{j(j + 1) - l(l + 1) + s(s + 1)}{2j(j + 1)}\right] \\
  &= \mu_\text{B} B m_j g_J,
\end{align}</math>
where the quantity in square brackets is the [[Landé g-factor]] <math>g_J</math> of the atom (<math>g_L = 1,</math> <math>g_S \approx 2</math>), and <math>m_j</math> is the ''z''&nbsp;component of the total angular momentum. 

For a single electron above filled shells, with <math>s = 1/2</math> and <math>j = l \pm s</math>, the Landé g-factor can be simplified to
<math display="block">
 g_J = 1 \pm \frac{g_S - 1}{2l + 1}.
</math>

Taking <math>V_\text{M}</math> to be the perturbation, the Zeeman correction to the energy is
<math display="block">
 E_\text{Z}^{(1)} = \langle nljm_j | H_\text{Z}^' | nljm_j \rangle
  = \langle V_\text{M} \rangle_\Psi = \mu_\text{B} g_J B_\text{ext} m_j.
</math>

===Example: Lyman-alpha transition in hydrogen===
The [[Lyman alpha|Lyman-alpha transition]] in [[hydrogen]] in the presence of the [[spin–orbit interaction]] involves the transitions <math>2\,^2\!P_{1/2} \to 1\,^2\!S_{1/2}</math> and <math>2\,^2\!P_{3/2} \to 1\,^2\!S_{1/2}.</math>

In the presence of an external magnetic field, the weak-field Zeeman effect splits the <math>1\,^2\!S_{1/2}</math> and <math>2\,^2\!P_{1/2}</math> levels into 2 states each (<math>m_j = +1/2, -1/2</math>) and the <math>2\,^2\!P_{3/2}</math> level into 4 states (<math>m_j = +3/2, +1/2, -1/2, -3/2</math>). The Landé g-factors for the three levels are
<math display="block">\begin{align}
 g_J &= 2   & &\text{for}\ 1\,^2\!S_{1/2}\ (j = 1/2, l = 0), \\
 g_J &= 2/3 & &\text{for}\ 2\,^2\!P_{1/2}\ (j = 1/2, l = 1), \\
 g_J &= 4/3 & &\text{for}\ 2\,^2\!P_{3/2}\ (j = 3/2, l = 1).
\end{align}</math>

[[Image:Zeeman p s doublet.svg|right|300px]]
Note in particular that the size of the energy splitting is different for the different orbitals because the ''g<sub>J</sub>'' values are different. Fine-structure splitting occurs even in the absence of a magnetic field, as it is due to spin–orbit coupling. Depicted on the right is the additional Zeeman splitting, which occurs in the presence of magnetic fields.

{| class="wikitable" style="text-align:center"
|+ Dipole-allowed Lyman-alpha transitions in the weak-field regime
! Initial state<br/> <math>(n = 2, l = 1)</math><br/> <math>|j, m_j\rangle</math>
! Final state<br/> <math>(n = 1, l = 0)</math><br/> <math>|j, m_j\rangle</math>
! Energy<br/> perturbation
|-
| <math>\left| \frac{1}{2}, \pm\frac{1}{2} \right\rangle</math>
| <math>\left| \frac{1}{2}, \pm\frac{1}{2} \right\rangle</math>
| <math>\mp\frac{2}{3} \mu_\text{B} B</math>
|-
| <math>\left| \frac{1}{2}, \pm\frac{1}{2} \right\rangle</math>
| <math>\left| \frac{1}{2}, \mp\frac{1}{2} \right\rangle</math>
| <math>\pm\frac{4}{3} \mu_\text{B} B</math>
|-
| <math>\left| \frac{3}{2}, \pm\frac{3}{2} \right\rangle</math>
| <math>\left| \frac{1}{2}, \pm\frac{1}{2} \right\rangle</math>
| <math>\pm \mu_{\rm B}B </math>
|-
| <math>\left| \frac{3}{2}, \pm\frac{1}{2} \right\rangle</math>
| <math>\left| \frac{1}{2}, \pm\frac{1}{2} \right\rangle</math>
| <math>\mp\frac{1}{3} \mu_\text{B} B</math>
|-
| <math>\left| \frac{3}{2}, \pm\frac{1}{2} \right\rangle</math>
| <math>\left| \frac{1}{2}, \mp\frac{1}{2} \right\rangle</math>
| <math>\pm\frac{5}{3} \mu_\text{B} B</math>
|}