[UPSC CSE 2006] Angular momentum, spin etc (Q329, 20M)

(i) Using Pauli matrices \sigma_x, \sigma_y and \sigma_z, show that \left(\vec{\sigma} \cdot \vec{r}\right) \left(\vec{\sigma} \cdot \vec{p}\right) = \vec{r} \cdot \vec{p} + i \vec{\sigma} \cdot \vec{L} \hfill 10 (ii) For the radiation of wavelength 6000~\text{\AA}, determine the wavelength separation between its two component lines which are observed in the normal Zeeman effect. The magnetic field used is \frac{\pi}{4}~\text{weber}/\text{m}^2 and the specific charge of the electron = 1.76 \times 10^{11}~\text{C}~\text{kg}^{-1}. \hfill 10