Problems

1. Using the explicit forms for the $2\times 2$ Pauli matrices, verify the commutation (square brackets) and anti-commutation (braces) relations

   
   

   $$\left[ \sigma_i, \sigma_j \right] = 2i \epsilon_{ijk} \sigma... ... \quad \left\{ \sigma_i, \sigma_j \right\} = 2 \delta_{ij} I ,$$
   
   

   where $I$ is the $2\times 2$ unit matrix. Hence show that

   
   

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

2. Use the proven identity in the previously question to prove the result

   
   

   $$(\vec{\sigma} \cdot \vec{a}) (\vec{\sigma} \cdot \vec{b}) = ... ...\cdot \vec{b} I + i\vec{\sigma} \cdot \vec{a} \times \vec{b} .$$
   
   

   Using the explicit $2\times 2$ form for

   
   

   $$\vec{\sigma} \cdot \vec{p} = \left( \begin{array}{cc} p_z & p_x-ip_y \\ p_x+ip_y & -p_z \end{array} \right) ,$$
   
   

   show that

   
   

   $$(\vec{\sigma} \cdot \vec{p})^2 = \vec{p}^{\ 2} I .$$