Quantum Mechanics · Chapter 09

Angular momentum

The generators of rotations, and the three commutation relations that decide everything about them. Ladder operators, the quantum numbers $j$ and $m$ with the half-integers they allow, the matrices of angular momentum, the orbital case in which only whole numbers survive, spherical harmonics, and what a rotation does to a state of given $j$.

[J^_{x}, J^_{y}] = i ℏ J^_{z}

J^^{2} ∣ j m ⟩ = ℏ^{2} j (j + 1) ∣ j m ⟩

J^_{\pm} = J^_{x} \pm i J^_{y}

J^_{z} ∣ j m ⟩ = ℏ m ∣ j m ⟩

Lessons

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05Angular momentum matricesOnce the states $\ket{j,m}$ are known, every angular momentum operator is a matrix in them. $\hat{J}_{z}$ is diagonal, with $m\hbar$ down the diagonal. $\hat{J}_{+}$ and $\hat{J}_{-}$ sit just above and just below it, with square roots that come from the lengths of raised states. $\hat{J}_{x}$ and $\hat{J}_{y}$ are their halves and differences. For $j=\tfrac12$ the matrices are two by two and turn out to be the Pauli matrices of the spin chapter; for $j=1$ they are three by three.Read lesson →

06Orbital angular momentumFor 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.Read lesson →

07Spherical harmonicsThe 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.Read lesson →

08Rotating angular momentum statesRotate a state of definite $j$ and it stays within the same $j$: rotations commute with $\hat{J}^{2}$, so they only reshuffle the $2j+1$ states of one ladder among themselves. The reshuffling is a $(2j+1)\times(2j+1)$ unitary matrix, the rotation matrix. About the $z$ axis it is diagonal — each state just acquires a phase $e^{-im\varphi}$. About another axis it mixes the $m$ values, with probabilities that are fixed functions of the angle. And for half-integer $j$ a whole turn gives $-1$, a sign no ordinary object shows.Read lesson →

09Chapter practiceBring the chapter together: the operators $\hat{\mathbf{L}}=\hat{\mathbf{r}}\times\hat{\mathbf{p}}$, the commutation relations, the ladder operators, the spectrum of $j$ and $m$, the matrices of angular momentum, orbital angular momentum and its whole numbers, spherical harmonics and rotation matrices. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Angular momentum

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