Nothing here is solved. Four wave functions are written down and checked, and the checking is what establishes which shapes go with which quantum numbers.

§8.1 asserted the fuzzy-vector rules and §8.2 showed they are real. Neither said what any of it has to do with a wave function. This section closes that gap, and it does so in an unusually concrete way: rather than solving an equation, Phillips writes down four simple wave functions and applies the operators to them, reading off the angular momentum properties by inspection.

Classical first

For a particle at $\mathbf r = (x,y,z)$ with momentum $\mathbf p = (p_x,p_y,p_z)$, the orbital angular momentum about the origin is

with components $L_x = yp_z - zp_y$, $L_y = zp_x - xp_z$, $L_z = xp_y - yp_x$ and magnitude $|\mathbf L| = \sqrt{L_x^2 + L_y^2 + L_z^2}$.

The quantum operator

Replace $\mathbf p$ by $\hat{\mathbf p} = -i\hbar\nabla$ and you have the operator:

a vector operator with three components:

When $\Psi$ is known, expectation values follow the usual recipe: $\langle L_x\rangle = \int\Psi^*\hat L_x\Psi\,\mathrm d^3\mathbf r$ and $\langle L_x^2\rangle = \int\Psi^*\hat L_x^2\Psi\,\mathrm d^3\mathbf r$. And if $\Psi$ happens to be an eigenfunction, $\hat L_x\Psi = L_x\Psi$, then §4.3’s argument gives $\langle L_x\rangle = L_x$ and $\langle L_x^2\rangle = L_x^2$, so

— the state has a precise $x$ component. This is the machinery; the rest of the section is four examples.

Four wave functions, worked by inspection

The book’s strategy is worth naming, because it is unusual. Instead of solving an eigenvalue equation, it guesses four wave functions with simple angular dependence and applies the operators to see what comes out:

where $R(r)$ is any well-behaved function of $r = \sqrt{x^2+y^2+z^2}$. The labels $(0,0)$ and $(1,\pm1)$ are given in advance; the point of the section is to earn them.

The other two, and a warning about z

Replacing $z$ by $x$ or $y$ in $\psi_{(1,0)}$ gives

Both still have $|L| = \sqrt2\,\hbar$; but $\psi'$ has $L_x = 0$ with $L_y$ and $L_z$ uncertain, and $\psi''$ has $L_y = 0$ with $L_z$ and $L_x$ uncertain.

Finally, the two complex ones. Applying the operators to $x \pm iy$ gives

so both have $l = 1$ with $m_l = \pm1$ — same magnitude, opposite $z$ component — and neither is an eigenfunction of $\hat L_x$ or $\hat L_y$.

What the shapes look like

A particle with definite angular momentum has a wave function with a definite angular shape, and the probability density on a sphere is $|Y_{l,m_l}|^2$ — the book’s Figs. 8.4, 8.6 and 8.7. They are the same picture at $l = 1, 2, 3$, so here they are as one object:

Three properties, now earned

The section closes by collecting what these examples have shown:

• Orbital angular momentum is quantized, with $\hbar = 1.055\times10^{-34}$ J s as the natural unit. Every eigenvalue found above was $0$, $\pm\hbar$ or $2\hbar^2$ — never anything in between.
• It is at best a fuzzy vector fuzzy vector The chapter's name for a quantum angular momentum: an object with a definite magnitude and a definite value for exactly one Cartesian component, the other two being uncertain but quantized when measured. The classical picture of a vector pointing in a definite direction has no quantum counterpart. defined in ch. 8 — open in glossary . In every example, only the magnitude and one component could be specified — because the components are non-compatible observables in the sense of chapter 7.
• Definite angular momentum means a definite angular shape. Zero angular momentum gives a spherically symmetric wave function; non-zero gives angular dependence.