Griffiths Introduction to Elementary Particles · Problem 4.20

Bound States — David J. Griffiths

Physics — QFT · pauli-matrices-commutator-anticommutator

Problem

Use the results of Problem 4.19 to show that (a) The commutator,

[A, B] = AB - BA

, of two Pauli matrices is

[\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k

. (b) The anticommutator,

\{A, B\} = AB + BA

, is

\{\sigma_i, \sigma_j\} = 2\delta_{ij}

. (c) For any two vectors

\mathbf{a}

and

\mathbf{b}

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