Hydrogen electron transition intensity 2p - 1s

[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 ω_fi⁴ 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) e^i*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

I_Φ = ∫ e^i*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 !

 

[DrClaude]

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.