A He ion is a single electron bound to a helium nucleus of charge
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Quantum Mechanics · Origins of quantum theory
Action and Symmetry, Ch Hamilton-Jacobi theory and action-angle variables
Problem
A He+ ion is a single electron bound to a helium nucleus of charge +2e. Treat the nucleus as infinitely heavy, so the reduced mass is μ=me, and take k=Ze2/4πε0 with Z=2 in the Kepler Hamiltonian H=−μk2/2(Ir+Iθ+Iϕ)2. For an infinitely heavy proton, hydrogen's Bohr-Sommerfeld ground-state energy is −13.606 eV.
((a)) Explain how E1 scales with Z at fixed nq=1.
((b)) Hence compute the ground-state energy of He+.