The Exercise

An electron bound in a central potential has orbital states $|\ell, m\rangle$ whose angular wavefunctions are the spherical harmonics $Y_{\ell m}(\theta, \phi)$, with $\ell \geq 0$ the orbital and $m$ the magnetic quantum number. Light drives transitions between these states through the electric dipole moment $-e\vec{r}$ of the electron, with $e$ the elementary charge and $\vec{r}$ its position. In terms of $r = |\vec{r}|$, the components $z$ and $x \pm iy$ of the position are, up to constants, $r Y_{10}$ and $r Y_{1,\pm 1}$. The matrix element of the dipole between $|\ell_2, m_2\rangle$ and $|\ell_1, m_1\rangle$ is then a radial integral times an angular one, the integral over the sphere of $Y^*_{\ell_1 m_1} Y_{1m} Y_{\ell_2 m_2}$.

The electric dipole selection rules say when the angular integral can be nonzero: only if $\ell$ changes by one and, for the components $x \pm iy$, only if $m$ changes by one as well.

Spherical Harmonics

For an integer $\ell \geq 0$ and an integer $m$, the normalization constant $N_{\ell m}$ of the spherical harmonic $Y_{\ell m}$ is the real number $$N_{\ell m} = \frac{(-1)^{|m|}}{2^\ell\, \ell!} \sqrt{\frac{2\ell + 1}{4\pi}\, \frac{(\ell - |m|)!}{(\ell + |m|)!}} .$$(1)^✦

The polar part $\Theta_{\ell m}$ of $Y_{\ell m}$ is the real function of the polar angle $\theta$ $$\Theta_{\ell m}(\theta) = N_{\ell m}\, \sin^{|m|}\theta\, \left. \frac{d^{\ell + |m|}}{dx^{\ell + |m|}} \bigl(x^2 - 1\bigr)^\ell \right|_{x = \cos\theta} .$$(2)^✦

Multiplying by the azimuthal phase gives the spherical harmonic $Y_{\ell m}$, the complex function of the polar angle $\theta$ and the azimuth $\phi$ $$Y_{\ell m}(\theta, \phi) = \Theta_{\ell m}(\theta)\, e^{i m \phi} .$$(3)^✦

The Dipole Overlap

For integers $\ell_1, \ell_2 \geq 0$ and $m_1, m, m_2$, the dipole overlap $\langle \ell_1 m_1 | Y_{1m} | \ell_2 m_2 \rangle$ is the complex number $$\langle \ell_1 m_1 | Y_{1m} | \ell_2 m_2 \rangle = \int_0^\pi \left( \int_0^{2\pi} Y^*_{\ell_1 m_1}(\theta, \phi)\, Y_{1m}(\theta, \phi)\, Y_{\ell_2 m_2}(\theta, \phi)\, d\phi \right) \sin\theta\, d\theta ,$$(4)^✦ the angular integral of the exercise, taken with the surface element $d\Omega = \sin\theta\, d\theta\, d\phi$.

The Claim

What Is Checked

The claim is stated over the dipole overlap (4) of the spherical harmonics (3), built from their polar parts (2) and normalization constants (1). What the library proves about these constructions is the pair of selection rules: (5) for every integer $m$, and (6) for $m = \pm 1$.