Editing Selection rule (section)

==== Summary table ====
<math>J = L + S</math> is the total angular momentum, 
<math>L</math> is the [[azimuthal quantum number]], 
<math>S</math> is the [[spin quantum number]], and
<math>M_J</math> is the [[Total angular momentum quantum number|secondary total angular momentum quantum number]].
Which transitions are allowed is based on the [[hydrogen-like atom]]. The symbol <math>\not\leftrightarrow</math> is used to indicate a forbidden transition.
{| class="wikitable" style="text-align:center"
|-
! colspan=2 | Allowed transitions
! Electric dipole (E1)
! Magnetic dipole (M1)
! Electric quadrupole (E2)
! Magnetic quadrupole (M2)
! Electric octupole (E3)
! Magnetic octupole (M3)
|-
! rowspan=3 | Rigorous rules 	
! (1)
| colspan=2 | <math>\begin{matrix} \Delta J = 0, \pm 1 \\ (J = 0 \not \leftrightarrow 0)\end{matrix}</math>
| colspan=2 | <math>\begin{matrix} \Delta J = 0, \pm 1, \pm 2 \\ (J = 0 \not \leftrightarrow 0, 1;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2}\end{matrix})\end{matrix}</math>
| colspan=2 | <math>\begin{matrix}\Delta J = 0, \pm1, \pm2, \pm 3 \\ (0 \not \leftrightarrow 0, 1, 2;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2} \end{matrix}, \begin{matrix}{3 \over 2}\end{matrix};\ 1 \not \leftrightarrow 1) \end{matrix}</math>
|-
! (2)
| colspan=2 | <math>\Delta M_J = 0, \pm 1 \  (M_J = 0 \not \leftrightarrow 0</math> if <math>\Delta J=0)</math>
| colspan=2 | <math>\Delta M_J = 0, \pm 1, \pm2</math>
| colspan=2 | <math>\Delta M_J = 0, \pm 1, \pm2, \pm 3</math>
|-
! (3)
| <math>\pi_\text{f} = -\pi_\text{i}</math>
| colspan=2 | <math>\pi_\text{f} = \pi_\text{i}</math>	
| colspan=2 | <math>\pi_\text{f} = -\pi_\text{i}</math>
| <math>\pi_\text{f} = \pi_\text{i}</math>
|-
! rowspan=2 | LS coupling
! (4)
| One-electron jump<br/><math>\Delta L = \pm 1</math>
| No electron jump<br/><math>\Delta L = 0</math>,<br><math>\Delta n = 0</math>
| None or one-electron jump<br/><math>\Delta L = 0, \pm 2</math>
| One-electron jump<br/><math>\Delta L = \pm 1</math>
| One-electron jump<br/><math>\Delta L = \pm 1, \pm 3</math>
| One-electron jump<br/><math>\Delta L = 0, \pm 2</math>
|-	
! (5)
| If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math>
| If <math>\Delta S = 0</math>:<br/><math>\Delta L = 0\,</math>
| colspan=2 | If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \pm 2 \\ (L = 0 \not \leftrightarrow 0, 1)\end{matrix}</math>
| colspan=2 | If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \pm 2, \pm 3 \\ (L=0 \not \leftrightarrow 0, 1, 2;\ 1 \not \leftrightarrow 1)\end{matrix}</math>
|-
! Intermediate coupling
! (6)
| colspan=2 | If <math>\Delta S = \pm 1</math>:<br/><math>\Delta L = 0, \pm 1, \pm 2\,</math>
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2, 
\pm 3 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math>
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math> 	
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2, \pm 3, \pm 4 \\ (L = 0 \not \leftrightarrow 0, 1)\end{matrix}</math> 
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math> 
|}
In [[hyperfine structure]], the total angular momentum of the atom is <math>F = I + J,</math> where <math>I</math> is the [[Quantum number#Nuclear angular momentum quantum numbers|nuclear spin angular momentum]] and <math>J</math> is the total angular momentum of the electron(s). Since <math>F = I + J</math> has a similar mathematical form as <math>J = L + S,</math> it obeys a selection rule table similar to the table above.