1. It is lmown that the three-dimensional central field problem in quantum mechanics can be reduced to a one-dimensional problem in the presence of an effective potential energy that additionally contains what us called a centrifugal energy term. For the attractive Coulomb problem (and employing the atomic units) the effective potential energy has the form !( 4 l) 23 where l is the angular momentum and Z is the charge of the nucleus. For i nite this function has a minimum at some i" = r\". thus implying that the lowlying part of the spectrum Eur]. can be approximately evaluated by treating it as if it would be a linear oscillator problem whose spectrum is well-known. Employing this idea. i.e. expanding [1) up to the second order in r r\". compute the Coulomb spectrum and compare with the exact result 22 EH ,2 _ _. (2) F 2mr+t+1F You are going to find out that this approximation can be justified provided 5333*} . Therefore it is admissible from the outset to approximate 12 . r(r+1)s(r+5) 2,1; s=r+1ra {3) To facilitate the comparison between the exact and approximate treatments. present your results in terms of an expansion in powers of {Hr + H 2)}31 s: l which is the more accurate than his} range of applicability of the approximation