Selection Rules

Selection Rules#

Recall that every emission rate requires the calculation of the matrix element $⟨ ψ_{b} | r | ψ_{a} ⟩$

The selection rule is defined for the Hydrogen wavefunction where certain combinations of $n, l, m$ will result the matrix element to zero. Not only is this convenient for math but it means there are certain transitions that are not allowed.

⟨ n^{'} l^{'} m^{'} | r | n l m ⟩

No transition occurs unless $Δ m = \pm 1$ or $0$
No transition occur unless $Δ l = \pm 1$

Case $Δ m$ = 0, thus $m = m^{'}$ : The matrix element only survives in $z$

⟨ n^{'} l^{'} m^{'} | r | n l m ⟩ = ⟨ n^{'} l^{'} m | z | n l m ⟩ \hat{z}

The length square of these matrix element vector is then

\bigabs {⟨ n^{'} l^{'} m^{'} | r | n l m ⟩}^{2} = \bigabs {⟨ n^{'} l^{'} m | z | n l m ⟩}^{2}

Case $Δ m = \pm 1$ thus $(m^{'} - m)^{2} = 1$ : The matrix element only survives in $x$ and $y$ survives such that

\begin{array}{r} ⟨ n^{'} l^{'} m \pm 1 | r | n l m ⟩ = ⟨ n^{'} l^{'} m \pm 1 | x | n l m ⟩ + ⟨ n^{'} l^{'} m \pm 1 | y | n l m ⟩ \\ \pm ⟨ n^{'} l^{'} m \pm 1 | x | n l m ⟩ = i ⟨ n^{'} l^{'} m | y | n l m ⟩ \\ ⟨ n^{'} l^{'} m \pm 1 | r | n l m ⟩ = ⟨ n^{'} l^{'} m \pm 1 | x | n l m ⟩ (\hat{x} + \mp i \hat{y}) \\ \abs {⟨ n^{'} l^{'} m \pm 1 | r | n l m ⟩}^{2} = 2 \bigabs {⟨ n^{'} l^{'} m \pm 1 | x | n l m ⟩}^{2} \end{array}