Quantum Mechanics · Chapter 16

Adding angular momenta

Two angular momenta in one system: the tensor product their states live in, the coupled and uncoupled bases, the rule that says which totals are possible, singlet and triplet, Clebsch–Gordan coefficients, spin–orbit coupling, the selection rules that decide which transitions light can drive, and the Wigner–Eckart theorem that says why they hold.

H = H_{A} \otimes H_{B}

J^= J^_{1} + J^_{2}

J = ∣ j_{1} - j_{2} ∣, \dots, j_{1} + j_{2}

∣ 0, 0 ⟩ = \frac{∣ ↑↓ ⟩ - ∣ ↓↑ ⟩}{2}

Lessons

Build the foundation

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04Two spins: singlet and tripletTwo spins $\tfrac12$ make totals 0 and 1: three states with $S=1$, the triplet, and one with $S=0$, the singlet.Read lesson →

05Clebsch–Gordan coefficientsThe numbers that write each coupled state in the product basis are the Clebsch–Gordan coefficients. They are found the way the triplet was: start at the top state, the one product with the largest $M$; lower it with the total lowering operator to fill the multiplet of the largest $J$; at the next value of $M$ the state orthogonal to what has been built is the top of the next multiplet; and so on down. With the standard choice of phases they are real numbers. They vanish unless $M=m_{1}+m_{2}$ and the triangle rule holds, and their squares are probabilities: the chance of finding each pair $(m_{1},m_{2})$ in a state of definite $J$.Read lesson →

06Spin–orbit couplingAn electron moving through the electric field of a nucleus sees, in its own rest frame, a magnetic field, and its magnetic moment feels it: an energy proportional to $\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}$. This spin–orbit interaction does not commute with $\hat{L}_{z}$ or $\hat{S}_{z}$, since it turns orbit and spin into each other, but it commutes with $\hat{L}^{2}$, $\hat{S}^{2}$ and the total $\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$. So its eigenstates are the coupled states, and a level with $\ell>0$ splits in two, $j=\ell+\tfrac12$ and $j=\ell-\tfrac12$. The yellow light of sodium is split this way into two lines 0.6 nm apart.Read lesson →

07Selection rulesLight drives an atom from one level to another through the electric dipole operator $e\hat{\mathbf{r}}$: a jump from $\ket{i}$ to $\ket{f}$ goes at a rate proportional to $|\bra{f}\hat{\mathbf{r}}\ket{i}|^{2}$. Most of these matrix elements vanish for reasons of symmetry alone. Parity: $\hat{\mathbf{r}}$ is odd, so the two levels must have opposite parity. Angular momentum: $\hat{\mathbf{r}}$ is a vector, which carries one unit, so $\ell$ can change by at most one — and with parity, by exactly one — while $m$ changes by 0 or $\pm1$. These are the dipole selection rules. In hydrogen $2p\to1s$ is allowed and $2s\to1s$ is forbidden, and 2s lives about a hundred million times longer as a result.Read lesson →

08The Wigner–Eckart theoremThe selection rules are special cases of one theorem. Operators, like states, come in families that rotate into one another: a scalar is unchanged by rotations, a vector’s three components mix like the three states of $j=1$, and a tensor operator of rank $k$ has $2k+1$ components that rotate like the states $\ket{k,q}$. The Wigner–Eckart theorem says that all the matrix elements of such an operator between two multiplets are one number, the reduced matrix element, times a Clebsch–Gordan coefficient. The geometry — which $m$, $m'$ and $q$ — is in the coefficient; the physics — how strong — is in the one number.Read lesson →

09Chapter practiceBring the chapter together: tensor products, the two bases, the addition rule, singlet and triplet, Clebsch–Gordan coefficients, spin–orbit coupling, the selection rules and the Wigner–Eckart theorem. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Adding angular momenta

Build the foundation