[UPSC CSE 2009] Schrodinger’s Wave equation and applications (Q201, 20M)

(i) Consider a particle in a three-dimensional box. Derive an expression for g(E), the density of states. (ii) Show that \frac{g(p)}{g(E)} = \frac{dE}{dp} , where g(p) is the density of states in the momentum space. Deduce that g(p) is proportional to p^2 for a free non-relativistic particle.