Editing Zeeman effect (section)

==Strong field (Paschen–Back effect)==
The Paschen–Back effect is the splitting of atomic energy levels in the presence of a strong magnetic field. This occurs when an external magnetic field is sufficiently strong to disrupt the coupling between orbital (<math>\vec{L}</math>) and spin (<math>\vec{S}</math>) angular momenta. This effect is the strong-field limit of the Zeeman effect. When <math>s = 0</math>, the two effects are equivalent. The effect was named after the German physicists [[Friedrich Paschen]] and [[Ernst Emil Alexander Back|Ernst E. A. Back]].<ref>{{cite journal |last1=Paschen |first1=F. |last2=Back |first2=E. |title=Liniengruppen magnetisch vervollständigt |journal=Physica |date=1921 |volume=1 |pages=261–273 |trans-title=Line groups magnetically completed [i.e., completely resolved] |language=German}} Available at: [https://www.lorentz.leidenuniv.nl/history/proefschriften/Physica/Physica_1_1921_05391.pdf Leiden University (Netherlands)]</ref>

When the magnetic-field perturbation significantly exceeds the spin–orbit interaction, one can safely assume <math>[H_{0}, S] = 0</math>. This allows the expectation values of <math>L_{z}</math> and <math>S_{z}</math> to be easily evaluated for a state <math>|\psi\rangle </math>. The energies are simply

:<math> E_{z} = \left\langle \psi \left| H_{0} + \frac{B_{z}\mu_{\rm B}}{\hbar}(L_{z}+g_{s}S_z) \right|\psi\right\rangle = E_{0} + B_z\mu_{\rm B} (m_l + g_{s}m_s). </math>

The above may be read as implying that the LS-coupling is completely broken by the external field. However, <math>m_l</math> and <math>m_s</math> are still "good" quantum numbers. Together with the [[selection rule]]s for an [[electric dipole transition]], i.e., <math>\Delta s = 0, \Delta m_s = 0, \Delta l = \pm 1, \Delta m_l = 0, \pm 1</math> this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the <math>\Delta m_l = 0, \pm 1</math> selection rule. The splitting <math>\Delta E = B \mu_{\rm B} \Delta m_l</math> is ''independent'' of the unperturbed energies and electronic configurations of the levels being considered.

More precisely, if <math>s \ne 0</math>, each of these three components is actually a group of several transitions due to the residual spin–orbit coupling and relativistic corrections (which are of the same order, known as 'fine structure'). The first-order perturbation theory with these corrections yields the following formula for the hydrogen atom in the Paschen&ndash;Back limit:<ref>{{cite book |author=Griffiths, David J. |title=Introduction to Quantum Mechanics |date=2004 |publisher=[[Prentice Hall]] |isbn=0-13-111892-7 |edition=2nd |page=280 |oclc=40251748}}</ref>
:<math> E_{z+fs} = E_{z} + \frac{m_e c^2 \alpha^4}{2 n^3} \left\{ \frac{3}{4n} - \left[ \frac{l(l+1) - m_l m_s}{l(l+1/2)(l+1) } \right]\right\}.</math>

===Example: Lyman-alpha transition in hydrogen===
In this example, the fine-structure corrections are ignored.

{| class="wikitable"
|+Dipole-allowed Lyman-alpha transitions in the strong-field regime
!Initial state
(<math>n=2,l=1</math>)

<math>\mid m_l, m_{s}\rangle</math>
!Initial energy perturbation
!Final state
(<math>n=1,l=0</math>)

<math>\mid m_l, m_{s}\rangle</math>
!Final energy perturbation
|-
|<math>\left| 1, \frac{1}{2}\right\rangle</math>
|<math>+2\mu_{\rm B}B_{z}</math>
|<math>\left| 0, \frac{1}{2}\right\rangle</math>
|<math>+\mu_{\rm B}B_{z}</math>
|-
|<math>\left| 0, \frac{1}{2}\right\rangle</math>
|<math>+\mu_{\rm B}B_{z} </math>
|<math>\left| 0, \frac{1}{2}\right\rangle</math>
|<math>+\mu_{\rm B}B_{z}</math>
|-
|<math>\left| 1, -\frac{1}{2}\right\rangle</math>
|<math>0 </math>
|<math>\left| 0, -\frac{1}{2}\right\rangle</math>
|<math>-\mu_{\rm B}B_{z}</math>
|-
|<math>\left| -1, \frac{1}{2}\right\rangle</math>
|<math>0 </math>
|<math>\left| 0, \frac{1}{2}\right\rangle</math>
|<math>+\mu_{\rm B}B_{z}</math>
|-
|<math>\left| 0, -\frac{1}{2}\right\rangle</math>
|<math>-\mu_{\rm B}B_{z} </math>
|<math>\left| 0, -\frac{1}{2}\right\rangle</math>
|<math>-\mu_{\rm B}B_{z}</math>
|-
|<math>\left| -1, -\frac{1}{2}\right\rangle</math>
|<math>-2\mu_{\rm B}B_{z} </math>
|<math>\left| 0, -\frac{1}{2}\right\rangle</math>
|<math>-\mu_{\rm B}B_{z}</math>
|}

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