# How is the ΔL = ±1 selection rule derived?

I understand there are selection rules in spectroscopy and quantum mechanics which governs whether a transition happens or not. For example, in atomic spectroscopy, the total orbital angular momentum quantum number $$L$$ can only change by plus or minus one:

$$\Delta L = \pm 1$$

I'd like to know the derivation of this selection rule, but I dont know where to start from.

• I assume you mean atomic spectroscopy, but please edit if I have misinterpreted. May 26, 2020 at 18:15
• The photon is spin 1. Note that this is not an absolute rule. May 26, 2020 at 18:19
• yes i mean atomic spectroscopy, sorry! How could I continue on from this @JonCuster May 26, 2020 at 19:54
• It is due to the photon, whose spin is equal to $1$. If a photon is absorbed, this spin is not lost. It must change the quantum number of the target. May 26, 2020 at 20:36
• Angular momentum is always conserved. The photon has one unit and so the atoms's angular momentum must change by this amount also. When examining the equations, involving the dipole $E.x$ induced in the atom(or molecule) by the radiation and the wavefunctions, the selection rules are found so that a solution is not zero, i.e. there is a finite chance of absorbing a photon. In one dimension the equations have the form $E\int\psi_f x\psi_idx$ where $\psi_{i,j}$ are the initial and final wavefunctions and $x$ the displacement that causes the dipole, and $E$ the amplitude of the radiation. May 27, 2020 at 8:33