While I am reviewing my undergrad physics for qualifying exam, I became confused about the numerous selection rules. (1) We have the selection rules for spontaneous emission in a hydrogen atom: Δl=±1 ,Δml=0,±1. This rule is the easiest to understand by evaluating <n'l'm'|z|nlm> (2) We have the selection rules for Zeeman effect. Δj=0,±1 ,Δmj=0,±1. I don't understand the reasons for these rules. (3) We also have the selections rules for a helium atom where one of its electron is excited: (Modern Physics by Randy Harris, Section 8.9): Δl_total=±1, Δs_total=0, Δj_total=0,±1 (0->0 forbidden). I am totally no clue on this one. (4) What about a hydrogen atom, without external magnetic field, but in contrast to (1), we are now considering fine structure so the good quantum numbers are n,l,j,mj?