Someone proposes a model potential for a heavy hydrogenic ion of nuclear charge :…
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
Someone proposes a model potential for a heavy hydrogenic ion of nuclear charge Z:V(r)=−4πϵ0rZe2−r2b,where b>0 is a small constant representing a departure from a pure Coulomb field close to the nucleus.
((a)) State whether this potential leaves the Runge-Lenz vector conserved, and predict whether the n2-fold degeneracy in l (at fixed n) survives. Without solving the radial equation exactly, explain why states of different l at the same n should shift by different amounts.
((b)) The nucleus of iron-56 with a single electron left (hydrogenic Fe25+) has Z=26, and its measured ionisation energy is 9277.7eV. Evaluate Zα with α=1/137.036 and the scale (Zα)2 of what a non-relativistic treatment misses. Then compare the measured value with the Coulomb value Z2×13.6057eV (reduced mass ignored), decide whether Coulomb scaling from hydrogen is adequate to better than 1 per cent for this ion, and say what the comparison implies about deciding a 1 per cent question from (Zα)2 alone.