A Modern Approach to Quantum Mechanics · Problem 3.4

Time Evolution — John S. Townsend

Physics — Quantum Mechanics · pauli-spin-matrices

Problem

Verify that (a)

\boldsymbol{\sigma} \times \boldsymbol{\sigma} = 2i\boldsymbol{\sigma}

and (b)

\boldsymbol{\sigma} \cdot \mathbf{a}\, \boldsymbol{\sigma} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{b}\,\mathbb{I} + i\boldsymbol{\sigma} \cdot (\mathbf{a} \times \mathbf{b})

, where

\boldsymbol{\sigma} = \sigma_x\mathbf{i} + \sigma_y\mathbf{j} + \sigma_z\mathbf{k}