A Modern Approach to Quantum Mechanics · Problem 3.3

Time Evolution — John S. Townsend

Physics — Quantum Mechanics · pauli-spin-matrices

Problem

Show that the Pauli spin matrices satisfy

\sigma_i\sigma_j + \sigma_j\sigma_i = 2\delta_{ij}\mathbb{I}

, where

i

and

j

can take on the values 1, 2, and 3, with the understanding that

\sigma_1 = \sigma_x

\sigma_2 = \sigma_y

, and

\sigma_3 = \sigma_z

. Thus for

i = j

show that

\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = \mathbb{I}

, while for

i \neq j

show that

\{\sigma_i, \sigma_j\} = 0

, where the curly brackets are called an anticommutator, which is defined by the relationship

\{\hat{A}, \hat{B}\} \equiv \hat{A}\hat{B} + \hat{B}\hat{A}