[UPSC CSE 2009] Schrodinger’s Wave equation and applications (Q199, 20M)

(i) The quantum mechanical probability distribution function of an electron in the ground state of the hydrogen atom is P(r) = N r^2 \exp (-2br). Using the result \int_0^\infty P(r) \, dr = 1, deduce that N is proportional to b^3. (ii) Prove that the value of 40 \, k_B T at T = 300 \text{ K} is nearly 1 \text{ eV}. Hence determine the Fermi temperature of a metal whose Fermi energy is 9 \cdot 4 \text{ eV}. (iii) Show that the Fermi velocity is related to the Fermi energy of electrons through the relation \frac{v_F}{c} = 1 \cdot 98 \left( \frac{E_F}{1 \text{ MeV}} \right)^{1/2} .