Griffiths Introduction to Elementary Particles · Problem 4.19

Bound States — David J. Griffiths

Physics — QFT · pauli-matrices-algebra

Problem

(a) Show that

\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = 1

. (``1'' here really means the

2 \times 2

unit matrix; if no matrix is specified, the unit matrix is understood.) (b) Show that

\sigma_x \sigma_y = i\sigma_z

\sigma_y \sigma_z = i\sigma_x

\sigma_z \sigma_x = i\sigma_y

. These results are neatly summarized in the formula

\sigma_i \sigma_j = \delta_{ij} + i\epsilon_{ijk}\sigma_k

(summation over

k

implied), where

\delta_{ij}

is the Kronecker delta:

\delta_{ij} = \begin{cases} 1, & \text{if } i = j \\ 0, & \text{otherwise} \end{cases}

and

\epsilon_{ijk}

is the Levi-Civita symbol:

\epsilon_{ijk} = \begin{cases} 1, & \text{if } ijk = 123, 231, \text{ or } 312 \\ -1, & \text{if } ijk = 132, 213, \text{ or } 321 \\ 0, & \text{otherwise} \end{cases}