Quantum Mechanics in Three Dimensions — David J. Griffiths
Physics — Quantum Mechanics · hydrogen_numerical
Problem
Find a few of the Bohr energies for hydrogen by "wagging the dog" (Problem 2.55), starting with Equation 4.53—or, better yet, Equation 4.56; in fact, why not use Equation 4.68 to set ρ0=2n, and tweak n? We know that the correct solutions occur when n is a positive integer, so you might start with n=0.9,1.9,2.9, etc., and increase it in small increments—the tail should wag when you pass 1, 2, 3, .... Find the lowest three ns, to four significant digits, first for ℓ=0, and then for ℓ=1 and ℓ=2. Warning: Mathematica doesn't like to divide by zero, so you might change ρ to (ρ+0.000001) in the denominator. Note: u(0)=0 in all cases, but u′(0)=0 only for ℓ≥1 (Equation 4.59). So for ℓ=0 you can use u(0)=0, u′(0)=1. For ℓ>0 you might be tempted to use u(0)=0 and u′(0)=0, but Mathematica is lazy, and will go for the trivial solution u(ρ)≡0; better, therefore, to use (say) u(1)=1 and u′(0)=0.