Selection Rules

Recall that every emission rate requires the calculation of the matrix element \(\bra{\psi_b}\mathbf{r}\ket{\psi_a}\)

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.

\[ \bra{n'l'm'}\mathbf{r}\ket{nlm} \]

1. No transition occurs unless \(\Delta m = \pm 1\) or \(0\)

2. No transition occur unless \(\Delta l = \pm 1\)

Case \(\Delta m\) = 0, thus \(m = m'\) : The matrix element only survives in \(z\)

\[ \boxed{\bra{n'l'm'}\mathbf{r}\ket{nlm} = \bra{n'l'm}z\ket{nlm} \hat z}\]

The length square of these matrix element vector is then

\[ \boxed{\bigabs{\bra{n'l'm'}\mathbf{r}\ket{nlm}}^2 = \bigabs{\bra{n'l'm}z\ket{nlm}}^2}\]

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

\[\begin{split} \bra{n'l'm\pm 1}\mathbf{r}\ket{nlm} = \bra{n'l'm\pm 1}x\ket{nlm} + \bra{n'l'm \pm 1}y\ket{nlm}\\ \pm \bra{n'l'm\pm 1}x\ket{nlm} = i\bra{n'l'm}y\ket{nlm}\\ \boxed{\bra{n'l'm\pm 1}\mathbf{r}\ket{nlm} = \bra{n'l'm\pm 1}x\ket{nlm}\bigg(\hat x + \mp i \hat y \bigg)}\\ \boxed{\abs{\bra{n'l'm\pm 1}\mathbf{r}\ket{nlm}}^2 = 2\bigabs{\bra{n'l'm\pm 1}x\ket{nlm}}^2} \end{split}\]