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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 probabilty. Thus, where does lie the differense between the mathematical zero (vanishing integrals) & the very minute transition probabilty ? Is it a result of the due integrals being only approximate themselves ?