Quantum Mechanics · Lesson 09

Chapter practice

Bring 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.

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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}

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The Wigner–Eckart theorem

Intuition

Bring 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.

A navigator adds a ship’s heading and the current to find where it will land. Adding angular momenta is the same bookkeeping, done in whole steps.

Before you practise

$J$ runs from $|j_{1}-j_{2}|$ to $j_{1}+j_{2}$ ; $M=m_{1}+m_{2}$ ; $\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}=\tfrac12(\hat{J}^{2}-\hat{L}^{2}-\hat{S}^{2})$ ; dipole jumps need $\Delta\ell=\pm1$ and $\Delta m=0,\pm1$ .

\sum_{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}(2J+1)=(2j_{1}+1)(2j_{2}+1)

Reminders

Two spins $\tfrac12$ : a symmetric triplet and an antisymmetric singlet.
Clebsch–Gordan coefficients vanish unless $M=m_{1}+m_{2}$ and the triangle rule holds.
A tensor of rank $k$ changes $m$ by $q$ and $j$ by at most $k$ .

Applications