LS vs jj couplings and their selection rules

[bentzy]

Two questions, where the 1st is related to previous discussion regarding thes couplings:
The selection rules for LS coupling is quite clear - it's based on calculating the compatible electric dipole matrix element. However, in the case of jj coupling we end up with different selection rules, which implies that the appropriate calculation here is basd on some other matrix element. What is the right (or approximate) matrix element, if not the electric dipole's ?
My 2nd question addresses Math vs Physics regarding selection rules in quantum physics. These are based on calculating the respective electric dipole matrix element & finding out under what conditions we get non-zero results. However, physically, forbidden transitions aren't strictly zero, but rather of very low probability. Thus, where does lie the differense between the mathematical zero (vanishing integrals) & the very minute transition probability ? Is it a result of the due integrals being only approximate themselves ?

 

[PeterDonis]

For reference, the previous thread is here:

https://www.physicsforums.com/threads/vector-sum-schemes-for-ls-coupling-jj-coupling.968076/

 

[kuruman]

The basic principle behind selection rules is angular momentum conservation. A photon carries angular momentum $±1*\hbar$ if it's right/left circularly polarized and $0*\hbar$ if linearly polarized. Hence the selection rules of $\pm1,~0$

Forbidden transitions are not the result of inaccurate integral evaluation. They have more to do with one's conscious choice to choose one basis set of representation to represent matrix elements over another, e.g. LS coupling vs. jj coupling. If, for example, you choose a basis set appropriate to LS coupling to write down Hamiltonian matrix elements for a physical system, this does not mean that spin-orbit coupling is completely turned off. When you diagonalize the Hamiltonian to get the energy levels between which transitions may occur, you will not get pure states. There will be admixtures which will introduce non-zero transition probabilities to the (mostly) forbidden transitions.

 

[DrClaude]

Thus, where does lie the differense between the mathematical zero (vanishing integrals) & the very minute transition probability ?

It comes from the use of the electric dipole moment as the only coupling term. Forbidden transitions can be possible due to the electric quadrupole, magnetic dipole, etc.