# I Hydrogen electron transition intensity 2p - 1s

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1. Nov 27, 2016

### FrostInMyTeeth

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 !

2. Nov 27, 2016

### Staff: Mentor

You forgot the polarization of the photon. The term in the middle of the integral is not simply r, but $\mathbf{r} \cdot \mathbf{\epsilon}$. That will add an angular compnent, which can be expressed in terms of $Y_{1,0}$, $Y_{1,+1}$, or $Y_{1,-1}$. The angular part is then the integral of three spherical harmonics.