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For many-electron atoms, the dipole selection rules are $\Delta M_J=0,\pm 1$ and $\Delta J=0,\pm 1$ with the transition $J=0\to J^\prime=0$ absolutely forbidden. This remains true even for quadrupole transitions and magnetic dipole transitions.

Can this be proved or is it an empirical fact? Is $j=0\to j^\prime=0$ also forbidden for hydrogenic atoms?

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If $J_i$ is the angular momentum of the initial state and $L$ is the angular momentum of the $L$-pole radiation, then the final state must have angular momentum $J_f$ that is in the coupling of $J_i\otimes L$. Thus, for dipole $L=1$ transitions, we have $J_f=J_i+1, J_i$ or $J_i-1$ provided $J_i-1\ge 0$. For quadrupole $L=2$ transitions we have $J_i+2\le L_f \le \vert J_i-2\vert$.

Using this with $J_i=0$, one simply notes that $0\otimes L=L$, and is thus $J_f=0$ is never allowed.

The less mathematical explanation is that radiation from and $L$-pole transition carries $L$ units of angular momentum, so that, if you start with $J_i=0$, returning to $J_f=0$ when the radiation carries some angular momentum would violate conservation of angular momentum.

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  • $\begingroup$ By $\vec{J}$, I meant $\vec{J}=\vec{L}+\vec{S}$ not only $\vec{L}$. $\endgroup$ – mithusengupta123 Dec 8 '18 at 4:35
  • $\begingroup$ It makes no difference. Replace my $L_i$ and $L_f$ by $J_i$ and $J_f$: the angular momentum coupling rules are for the same for $J$ or $L$. $\endgroup$ – ZeroTheHero Dec 8 '18 at 4:37
  • $\begingroup$ "when the radiation carries some angular momentum would violate conservation of angular momentum." But $\Delta J=0$, for example, $J_i=1$ to $J_f=1$ then also violate conservation of angular momentum? $\endgroup$ – mithusengupta123 Dec 8 '18 at 13:15
  • $\begingroup$ @mithusengupta123 no. Angular momentum is a “vector” so it is quite possible to have the situation you describe, somewhat similar to hyperphysics.phy-astr.gsu.edu/hbase/Atomic/lcoup.html. $\endgroup$ – ZeroTheHero Dec 8 '18 at 13:21
  • $\begingroup$ @mithusengupta123 ... or slide 22 of this: slideshare.net/mobile/ibenk97/… which gives an example with $\ell=2$ and $\ell=1$, which would apply to dipole transitions from a $J_i=2$ state. $\endgroup$ – ZeroTheHero Dec 8 '18 at 13:29

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