Spherical harmonics

Intuition

The joint eigenfunctions of $\hat{L}^{2}$ and $\hat{L}_{z}$ are functions of direction alone, the spherical harmonics $Y_{\ell m}(\theta,\varphi)$ . Their $\varphi$ dependence is $e^{im\varphi}$ ; their $\theta$ dependence is a polynomial in $\cos\theta$ and $\sin\theta$ . The top one of each ladder is simply $\sin^{\ell}\theta e^{i\ell\varphi}$ , and lowering it gives the rest. Their shapes — a sphere for $\ell=0$ , two lobes for $\ell=1$ , lobes and a ring for $\ell=2$ — are the shapes chemists draw for s, p and d orbitals.

A globe can be coloured by latitude and longitude in patterns with a few bands and a few meridians; the spherical harmonics are the patterns with the fewest bands and meridians for each amount of wiggling, and every colouring of the globe is a sum of them.

Polar plots of $|Y_{10}|^{2}$ and $|Y_{20}|^{2}$ in a plane containing the axis, which points up: the distance from each centre in a direction is the probability density in that direction. has two lobes along ; has two lobes and a ring round the middle, separated by cones where it vanishes.

$Y_{\ell m}(\theta,\varphi)$ is the normalised joint eigenfunction of $\hat{L}^{2}$ and . It is orthonormal over the sphere, and every function of direction is a series in them.

\begin{aligned}Y_{00}&=\frac{1}{\sqrt{4\pi}},\qquad Y_{10}=\sqrt{\frac{3}{4\pi}}\cos\theta\\ Y_{1,\pm1}&=\mp\sqrt{\frac{3}{8\pi}}\sin\theta\,e^{\pm i\varphi}\end{aligned}

Properties

Orthonormality on the sphere: $\int Y_{\ell'm'}^{*}Y_{\ell m}\,d\Omega=\delta_{\ell\ell'}\delta_{mm'}$ with .

The top of each ladder

\hat{L}_{+}Y_{\ell\ell}=0\ \Longrightarrow\ Y_{\ell\ell}\propto\sin^{\ell}\theta\,e^{i\ell\varphi}

The top state has $m=\ell$ , so its $\varphi$ dependence is $e^{i\ell\varphi}$ , and raising it gives zero. With the raising operator in spherical coordinates, that condition is a first-order equation for the $\theta$ part, solved by a power of .

Proof steps

\hat{L}_{+}=\hbar e^{i\varphi}\Bigl(\frac{\partial}{\partial\theta}+i\cot\theta\,\frac{\partial}{\partial\varphi}\Bigr)

The raising operator in spherical coordinates, quoted without its derivation.

Y_{\ell\ell}=f(\theta)\,e^{i\ell\varphi}

An eigenfunction of $\hat{L}_{z}$ with $m=\ell$ .

\hat{L}_{+}Y_{\ell\ell}=\hbar e^{i(\ell+1)\varphi}\bigl(f'-\ell\cot\theta\,f\bigr)

The $\varphi$ derivative brings down $i\ell$ , and $i\cdot i\ell=-\ell$ .

f'=\ell\cot\theta\,f\ \Rightarrow\ f=C\sin^{\ell}\theta

Separate and integrate, since $(\ln\sin\theta)'=\cot\theta$ .

Y_{\ell m}\propto\hat{L}_{-}^{\,\ell-m}\bigl(\sin^{\ell}\theta\,e^{i\ell\varphi}\bigr)

Lowering $\ell-m$ times gives every other harmonic of the ladder.

Applications

Practice

The Constant Harmonic

For $\ell=0$ the harmonic is a constant, the same in every direction. Normalising it over the whole sphere, of solid angle $4\pi$ , gives $1/\sqrt{4\pi}$ .

Y_{00}=\frac{1}{\sqrt{4\pi}}

Try it

What is $Y_{00}$ ? Give three decimal places.