Orbital angular momentum

Intuition

For angular momentum built from motion, $\hat{\mathbf{L}}=\hat{\mathbf{r}}\times\hat{\mathbf{p}}$ , the ladder argument allows too much. A wavefunction is a single-valued function of position, so going once round the $z$ axis must bring it back to itself. Its dependence on the angle $\varphi$ is $e^{im\varphi}$ , and that returns to itself only if $m$ is a whole number. So orbital angular momentum has whole-number values of $j$ , written $\ell$ , and the half-integers belong to something else — spin.

A necklace of beads closes only if the pattern fits the circle a whole number of times. A pattern that repeats two and a half times around would leave a seam, and a wavefunction cannot have a seam.

The real part of $e^{im\varphi}$ drawn as a ripple around a circle, once round the $z$ axis. For $m=3$ the ripple closes on itself. For $m=\tfrac12$ it comes back to a different value where it started, which a single-valued wavefunction cannot do.

In spherical coordinates the orbital operators act on the angles alone. $\hat{L}_{z}=-i\hbar\,\partial_{\varphi}$ , and is times the angular part of the Laplacian.

\hat{L}^{2}=-\hbar^{2}\Bigl[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\Bigl(\sin\theta\frac{\partial}{\partial\theta}\Bigr)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}\Bigr]

Properties

The eigenfunctions of $\hat{L}_{z}$ are $e^{im\varphi}$ , with eigenvalue .

Orbital angular momentum is whole

\begin{aligned}&\psi(\varphi+2\pi)=\psi(\varphi)\ \Longrightarrow\ m\in\mathbb{Z}\\ &\Longrightarrow\ \ell\in\{0,1,2,\dots\}\end{aligned}

An eigenfunction of $\hat{L}_{z}=-i\hbar\,\partial_{\varphi}$ depends on through . Going once round the axis multiplies it by , which must be 1. So is whole, and since the of one differ by whole numbers and reach , is whole too.

Proof steps

-i\hbar\,\partial_{\varphi}\psi=m\hbar\,\psi\ \Rightarrow\ \psi=f(r,\theta)\,e^{im\varphi}

A first-order equation in $\varphi$ .

\psi(\varphi+2\pi)=e^{2\pi im}\,\psi(\varphi)

Go once round the $z$ axis.

e^{2\pi im}=1\ \Rightarrow\ m\in\mathbb{Z}

A wavefunction has one value at each point.

\ell=m_{\max}\in\{0,1,2,\dots\}

The largest $m$ is $\ell$ , and it is a whole number.

Applications

Practice

Once Round the Axis

Going once round the $z$ axis multiplies $e^{im\varphi}$ by $e^{2\pi im}$ . A wavefunction must come back to itself, so is a whole number.

e^{2\pi im}=1\ \Rightarrow\ m\in\mathbb{Z}

Try it

Why can orbital angular momentum not have half-integer values of $m$ ?