please solve all of them

Problem 5.37 An important quantity in many calculations is the density of states G(E): G(E) dE = number of states with energy between E and E +de. For a one-dimensional band structure, = 2 da G(E) DE 2 1/N a where dq/(2 /N a) counts the number of states in the range dq (see Equation 5.63), and the factor of 2 accounts for the fact that states with q and -q have the same energy. Therefore 1 1 1 - Na G(E) 1 \d E/dq| (a) Show that for a = 0 (a free particle) the density of states is given by 1 m Gfree (E) a V 2E (b) Find the density of states for a #0 by differentiating Equation 5.71 with respect to q to determine d E/dq. Note: Your answer should be written as a function of E only (well, and a, m, h, a, and N) and must not contain q (use k as a shorthand for 2m E/h, if you like). Problem 5.37 An important quantity in many calculations is the density of states G(E): G(E) dE = number of states with energy between E and E +de. For a one-dimensional band structure, = 2 da G(E) DE 2 1/N a where dq/(2 /N a) counts the number of states in the range dq (see Equation 5.63), and the factor of 2 accounts for the fact that states with q and -q have the same energy. Therefore 1 1 1 - Na G(E) 1 \d E/dq| (a) Show that for a = 0 (a free particle) the density of states is given by 1 m Gfree (E) a V 2E (b) Find the density of states for a #0 by differentiating Equation 5.71 with respect to q to determine d E/dq. Note: Your answer should be written as a function of E only (well, and a, m, h, a, and N) and must not contain q (use k as a shorthand for 2m E/h, if you like)