Question

7. Radial Potential Problem a) (6pts) Beginning with the radial part of the Schrödinger equation (18) make the change of variables u(r) R(r) and find (3pts) (19) = 2mdr2 identify the effective potential (3pts). b) (8pts) Given the radial potential (20) Sketch Vetr (r) (2pts) and find u(r) in terms of : rsa (21) (22) in the regions r < a, and r a (2pts). Set the constants (other than k and k) that arise in your solution using the behavior of u(r) as r0, 0o (2pts). Find the equation (23) with (24) h2 using the continuity relations at r - a (2pts). c) (6pts) Draw a plot of the equation you found in part b as a function of z (2pts). Is there a solution if 20 < π/2 (2pts)? What is the relation for the sum of E and Vo in terms of z, h, m, and a (2pts)?

Answer 1

a the Schr�dinger wave eqn d/dr(r^2 dR(r)/dr) - 2mr^2/h^2 [V(r) - E]R = l(l + 1)R Charge of variable rightarrow u(r) = r R(r) then R(r) = u(r)/r implies dR(r)/dr = [r du/dr - u]/r^2 implies d/dr[r^2 (dR/dr)] = r d^2 u/dr^2 implies - h^2/2m d^2 u/dr^2 = Eu - [V + h^2/2m l(l + 1)/r^2]u & this is called Radial eqn \r\n and the effective potential v_eff = v + h^2/2m l(l + 1)/r^2 (b) Given v(r) = { - v_0, r lessthanorequalto a 0, r greaterthanorequalto a using the radial wave eqn we can get for r lessthanorequalto a, u(r) = A sin kr where k = squareroot 2m(E + v_0)/h r lessthanorequalto a & for r greaterthanorequalto a, the sol of radial eqn, with l = 0, v = 0 and (for a bound state) E lessthanorequalto 0 implies d^2 u/dr^2 = -2m/h^2 Eu = k^2 u with k = squareroot - 2m E/h implies u(r) = Ce^kr + De^- kr But Ce^kr term blow up