Introduction to Quantum Mechanics · Problem 4.29

Spin matrices encode angular momentum through their commutation structure—the same algebraic rules that govern rotations in ordinary space also govern spin rotations in quantum mechanics. Part (a) asks you to verify that $[\hat{S}_j, \hat{S}_k] = i\hbar \epsilon_{jkl} \hat{S}_l$ by direct matrix multiplication; part (b) then reveals the elegant underlying pattern by showing that the Pauli matrices themselves (which are $\hat{S}$ up to factors of $\hbar/2$) satisfy a closed product rule that automatically generates both the identity and the commutation relations. This product rule is the key: it tells you exactly what happens when you multiply any two Pauli matrices, making the algebra transparent.

Spin matrices encode angular momentum through their commutation structure—the same algebraic rules that govern rotations in ordinary space also govern spin rotations in quantum mechanics. Part (a) asks you to verify that $[\hat{S}_j, \hat{S}_k] = i\hbar \epsilon_{jkl} \hat{S}_l$ by direct matrix multiplication; part (b) then reveals the elegant underlying pattern by showing that the Pauli matrices themselves (which are $\hat{S}$ up to factors of $\hbar/2$) satisfy a closed product rule that automatically generates both the identity and the commutation relations. This product rule is the key: it tells you exactly what happens when you multiply any two Pauli matrices, making the algebra transparent.