[UPSC CSE 2018] Schrodinger’s Wave equation and applications (Q158, 15M)

Show that for free electron gas, the density of states in three dimensions (3D) varies as E^{1/2}, and this dependence changes to E^0 for 2D (quantum well), E^{-1/2} for 1D (quantum wire) and \delta function for 0D (quantum dot).