Editing Zeeman effect (section)

==Theoretical presentation==
The total [[Hamiltonian (quantum mechanics)|Hamiltonian]] of an atom in a magnetic field is
<math display="block">
 H = H_0 + V_\text{M},
</math>
where <math>H_0</math> is the unperturbed Hamiltonian of the atom, and <math>V_\text{M}</math> is the [[Perturbation theory|perturbation]] due to the magnetic field:
<math display="block">
 V_\text{M} = -\vec{\mu} \cdot \vec{B},
</math>
where <math>\vec{\mu}</math> is the [[magnetic moment]] of the atom. The magnetic moment consists of the electronic and nuclear parts; however, the latter is many orders of magnitude smaller and will be neglected here. Therefore,
<math display="block">
 \vec{\mu} \approx -\frac{\mu_\text{B} g \vec{J}}{\hbar},
</math>
where <math>\mu_\text{B}</math> is the [[Bohr magneton]], <math>\vec{J}</math> is the total electronic [[angular momentum]], and <math>g</math> is the [[Landé g-factor]].

A more accurate approach is to take into account that the operator of the magnetic moment of an electron is a sum of the contributions of the [[angular momentum operator|orbital angular momentum]] <math>\vec L</math> and the [[angular momentum operator|spin angular momentum]] <math>\vec S</math>, with each multiplied by the appropriate [[gyromagnetic ratio]]:
<math display="block">
 \vec{\mu} = -\frac{\mu_\text{B} (g_l \vec{L} + g_s \vec{S})}{\hbar},
</math>
where <math>g_l = 1</math>, and <math>g_s \approx 2.0023193</math> (the [[anomalous magnetic dipole moment|anomalous gyromagnetic ratio]], deviating from 2 due to the effects of [[quantum electrodynamics]]). In the case of the [[LS coupling]], one can sum over all electrons in the atom:
<math display="block">
 g \vec{J} = \Big\langle\sum_i (g_l \vec{l}_i + g_s \vec{s}_i)\Big\rangle = \big\langle(g_l \vec{L} + g_s \vec{S})\big\rangle,
</math>
where <math>\vec{L}</math> and <math>\vec{S}</math> are the total spin momentum and spin of the atom, and averaging is done over a state with a given value of the total angular momentum.

If the interaction term <math>V_\text{M}</math> is small (less than the [[fine structure]]), it can be treated as a perturbation; this is the Zeeman effect proper. In the Paschen–Back effect, described below, <math>V_\text{M}</math> exceeds the [[LS coupling]] significantly (but is still small compared to <math>H_0</math>). In ultra-strong magnetic fields, the magnetic-field interaction may exceed <math>H_0</math>, in which case the atom can no longer exist in its normal meaning, and one talks about [[Landau level#Landau levels|Landau levels]] instead. There are intermediate cases that are more complex than these limit cases.