Introduction to Quantum Mechanics · Problem 7.19

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · angular_momentum_commutators

Problem

Evaluate the following commutators: (a)

[\mathbf{L} \cdot \mathbf{S}, \mathbf{L}]

, (b)

[\mathbf{L} \cdot \mathbf{S}, \mathbf{S}]

, (c)

[\mathbf{L} \cdot \mathbf{S}, \mathbf{J}]

, (d)

[\mathbf{L} \cdot \mathbf{S}, L^2]

, (e)

[\mathbf{L} \cdot \mathbf{S}, S^2]

, (f)

[\mathbf{L} \cdot \mathbf{S}, J^2]

. Hint:

\mathbf{L}

and

\mathbf{S}

satisfy the fundamental commutation relations for angular momentum (Equations 4.99 and 4.134), but they commute with each other.