Let's say we have a transition from state 2p to 1s of an hydrogen atom's electron. The intensity radiated by the electron is given by I = 4/3 ωfi4 e2/c3 |< f | r | i >|2. If we take the the | 2 1 1 > → | 1 0 0 > transition for example, we must compute the following integral : < f | r | i > = C1 ∫∫∫ e-r/a r sin(θ) ei*Φ * e-r/2a sin(theta) r2 sin(θ)dr dθ dΦ , r = [ 0→ ∞], θ = [0 → pi], Φ = [0 → 2π] Which can be separated in three integrals (depending on r, θ and Φ respectively). The integral in Φ is IΦ = ∫ ei*Φ dΦ = 0 , Φ = [0 → 2π] Which means that < f | r | i > = 0 and that the transition is not allowed. However in the literature (and logically), this transition is allowed. I'm not quite sure if it's a comprehension issue or a simple computation error. Thank you !