Add to the Coulomb potential a small attractive inverse-square term, V(r)=- Ze^2 4…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure
Deriving the Spectrum, Ch The exact Coulomb problem and Rydberg states
Problem
Add to the Coulomb potential a small attractive inverse-square term, V(r)=−4πϵ0rZe2−r2b,b>0, and measure its strength by the dimensionless number β=2μb/ℏ2.
((a)) Show that the radial equation for u=rR is the Coulomb radial equation with l(l+1) replaced by s(s+1)=l(l+1)−β, with β<41. Show that for l=0 both roots give a u that vanishes at r=0, and pick the root for which the kinetic energy, through ∫0∞u′2dr, stays finite.
((b)) Carry the series-termination argument through with s in place of l, and show that the bound energies are E=−Z2hcRM/(nr+s+1)2 with nr=0,1,2,…
((c)) To first order in β, show that the level that grows out of the Coulomb level n=nr+l+1 moves by ΔE=−Z2b/[a2n3(l+21)], and identify this as −b⟨r−2⟩.
((d)) Take β=0.02. For n=2, find the exact ratio E/E2 for l=0 and for l=1, where E2 is the unperturbed Coulomb energy, and compare each with its first-order value.