[UPSC CSE 2005] Schrodinger’s Wave equation and applications (Q216, 10+10M)

(i) The ground state wavefunction of a linear simple harmonic oscillator is \psi = A \exp \left( - \frac{\alpha^2 x^2}{2} \right) Calculate the constant A and the average values of x^2 and x. Given that \int_0^\infty e^{-x^2} = \frac{\pi^{1/2}}{2} (ii) How can the pure rotation spectrum of \text{H}_2 molecule be observed ? If the bond length of \text{H}_2 molecule is 0 \cdot 07417 \text{ nm}, what would be the spacing of lines in its spectrum ?