Question

Extra problem 2: A lot of the math below is in Griffith's with slightly different notation. I'm asking you to do this because this is probably the single most important quantum mechanics problem ever: this is the one that started it all! In class, we found that the dimensionless radial equation for the Hydrogen atom could be reduced to A. Simplify the equation for the limit ρ → 0 and find the corresponding limiting solution. This is a second order differential equation, so you should find two solutions. However, when you impose the appropriate boundary conditions, you should find that one of the two solutions can be rejected, leaving you with just one solution. Find it B. Now simplify the radial equation in the limit ρ → 00, and compute the corresponding limiting solution. As before, you should find two solutions, one of which can be rejected because of boundary conditions. I strongly suggest that you define k such that k2for the purposes of making your algebra look prettier C. You should have found u d+1 for ρ . 0 and u e-kρ for ρ defining a function u(p) such that u(p-d+le-ko(p) oo. We will "peel off" this behavior by C.1. Starting from this definition, compute u' and u". C.2. Plug your expressions for u' and u" into Schrodinger's equation to get a differential equation for v Your final expression should look like this D. We will try a power-series solution, i.e. set u(p) Σο o PP. Before plugging in, compute each of the terms individually, and massage the indices until all terms look like Σ070(something)ρα D1. Find ρυ". D2. Find 2(l +1)v'.

Answer 1

untom e an as ro As drates Hence EaD wednces t 2. Lu 2. ナD as

vani sves. Henre 4 oedrc es to, 2. n of which can wwitt en As ave ence ewe A=0 C .1 dl

La we get 2. dl dp

2. -2-么 2 2.

D-1 anew series dv ヲ d G dp do D. 2 어ㅋ