As is shown in the pic

1. Suppose that the coulomb energy for a certain atomic core is modified to Bh2 Ze2 V(r) = r (1) (a) Write down the Schrodinger equation for a system that consists of one heavy nucleus with the electric charge Ze and an electron with such a potential. (b) Derive the radial wave equation. Are / and m still good quantum numbers for this problem? Explain your reasoning. Show that in this case, the effective "angular momentum" is * = -: + v (1 +1/2)2 + 24B. (c) Solve your radial wave equation (short distance behavior, long distance behavior, recursion relationship for the polynomial part, truncation of the polynomial) and get the energy levels. (d) What is the degeneracy of each energy level? Is there any restriction on the value of B7 (e) The relativisite analog of the Schrodinger equation for a spin 0 electron (hence not appilcable to real electron, which obeys the Dirac equation) is the operator version of the relativistic energy expression vp?c' + m?c' + V(r) = E (2) or rather its squared form: petm'd = (E-V) (3) For the hydrogen-like problem, this becomes 2 (c'p' + (mc] ]? ) (x) = ( E+) (x) (4) Rearrange this to the following form: p2 Z*e2 2m (I) = Ev(x) (5) Express constants B, Z* and E* in terms of the original parameters. (f) Show that the energy spectrum of ENR = E - mc is a-mc ENR (6) 2(n, + 1/2 + V(1+1/2)3 - 2303)2 keeping only the leading order terms in ENR/(mc?). [Caution: One cannot get the right relativistic corrections this way as the electron spin is completely ignored. Dirac equation is needed for the right answer.]