Any angular shape at all is a sum of these. That is what the chapter has been working toward, and the table of harmonics is the basis the sum is taken in.

§8.3a checked four hand-picked wave functions and found their angular momentum by inspection. That leaves the obvious question unanswered: what are all the states with definite angular momentum? This section names them — the spherical harmonics spherical harmonic Y_{l,m_l}(θ,φ), the simultaneous eigenfunctions of L̂² and L̂_z, satisfying L̂²Y = l(l+1)ħ²Y and L̂_zY = m_lħY. They form a complete orthonormal set of angular shapes on the sphere. defined in ch. 8 — open in glossary — and then does something more useful still: shows that every wave function is a superposition of them, which is a Fourier series in disguise.

The complete set of angular shapes

All possible orbital angular momentum properties are described by the simultaneous eigenfunctions of $\hat L^2$ and $\hat L_z$ — a legitimate pair to ask for, since ch07 problem 3 proved $[\hat L^2,\hat L_z] = 0$. These are the spherical harmonics $Y_{l,m_l}(\theta,\phi)$, defined by

with $l = 0,1,2,\dots$ and $m_l = -l,\dots,+l$. They are orthogonal,

and normalized,

where $\mathrm d\Omega = \sin\theta\,\mathrm d\theta\,\mathrm d\phi$ is the solid angle, integrated over $\theta$ from $0$ to $\pi$ and $\phi$ from $0$ to $2\pi$.

Comparing with §8.3a’s hand-picked functions confirms the labels that section earned: $\psi_{(0,0)} \propto Y_{0,0}$, $\psi_{(1,0)} \propto Y_{1,0}$ and $\psi_{(1,\pm1)} \propto Y_{1,\pm1}$.

Every harmonic has the same simple dependence on $\phi$:

while the $\theta$ dependence grows more complicated as $l$ increases.

The angular shape of the probability density is $|Y_{l,m_l}(\theta,\phi)|^2$ — Figs. 8.4, 8.6 and 8.7 for $l = 1, 2, 3$. There is no dependence on $\phi$, whatever $l$ and $m_l$, while the $\theta$ dependence grows richer with $l$.

Every wave function is a superposition

Now the payoff. Suppress $r$ and $\theta$ for a moment and look only at the $\phi$ dependence. Any complex function $\psi(\phi)$ on $0 \le \phi \le 2\pi$ has a Fourier series $\psi(\phi) = \sum_n c_n e^{in\phi}$. Rewriting it in the notation of quantum mechanics:

Problem 5 shows the $Z_{m_l}$ are eigenfunctions of $\hat L_z$ with eigenvalues $m_l\hbar$. So Eq. (8.27) is the principle of linear superposition again — and $|c_{m_l}|^2$ is the probability that a measurement of $L_z$ returns $m_l\hbar$.

The same argument in both angles gives the general expansion:

and orthonormality — Eqs. (8.24) and (8.25) — inverts it:

The worked example

Take $\psi'_{(1,0)} = R(r)\,x/r$ from Eq. (8.22) — the state that is sharp about the $x$ axis. What happens if you measure $L^2$ and $L_z$?

Where this goes next

The expansion of Eq. (8.28) is not only bookkeeping. For a scattered particle, the coefficient $c_{l,m_l}(r)$ is called a partial wave partial wave The coefficient c_{l,m_l}(r) in the spherical-harmonic expansion of a scattered wave. Each decomposes into incoming and outgoing spherical waves, and scattering shifts the phase of the outgoing one — the 3-D generalization of §5.1's phase shift. defined in ch. 8 — open in glossary : it splits into an incoming and an outgoing spherical wave, and the effect of scattering is to shift the phase of the outgoing one.

§5.1b met exactly this in one dimension, where a single phase shift $\delta$ carried all the information about the scattering. In three dimensions there is one phase shift for each $l$ — and from that set of numbers the whole scattering cross-section follows.