In the Coulomb problem the reduced radial function is written

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The hydrogen atom

Problem

In the Coulomb problem the reduced radial function is written u(ρ)=ρl+1e−ρw(ρ)u(\rho) = \rho^{l+1}e^{-\rho}w(\rho), with ρ=κr\rho = \kappa r, and ww obeys ρw′′+2(l+1−ρ)w′+[ρ0−2(l+1)]w=0.\rho w'' + 2(l+1-\rho)w' + [\rho_0 - 2(l+1)]w = 0. Termination of the power series for ww requires ρ0=2n\rho_0 = 2n with n=nr+l+1n = n_r+l+1, and then κ=1/(na0)\kappa = 1/(na_0), with a0=0.0529a_0 = 0.0529 nm. Consider a state in which ww is a constant, that is nr=0n_r=0. The proton is fixed, the electron is non-relativistic and spin is ignored. ((a)) Show that a constant ww satisfies the equation only if ρ0=2(l+1)\rho_0 = 2(l+1), and hence that such a state has l=n−1l=n-1. ((b)) Write Rn,n−1(r)=u/rR_{n,n-1}(r) = u/r up to a real positive constant, as a function of rr, nn and a0a_0. ((c)) Show that the radial probability density P(r)=∣Rn,n−1∣2r2P(r) = |R_{n,n-1}|^2r^2 has its maximum at r=n2a0r = n^2a_0. ((d)) Evaluate this most probable radius for n=7n=7, in nm.

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