[UPSC CSE 2005] Atomic Physics (Q104, 10+10M)

(i) How many lines occur in the multiplet arising from ^{2}\text{P}_{3/2} \rightarrow {^{2}\text{S}_{1/2}} and ^{2}\text{P}_{1/2} \rightarrow {^{2}\text{S}_{1/2}} transitions of alkali metal atoms placed in a weak magnetic field, why ? (ii) The wavefunction of a particle confined in a cube of volume L^3 is given by \Psi (x,y,z) = \left( \frac{2}{L} \right)^{3/2} \sin \frac{\pi x}{L} \sin \frac{\pi y}{L} \sin \frac{\pi z}{L} Calculate the average values of p_x and p_x^2 in the region 0 < x < L.