An electron of reduced mass is bound to a nucleus of charge by the Coulomb potential
PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Central potentials and canonical three-dimensional systems
The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem
Problem
An electron of reduced mass μ is bound to a nucleus of charge +Ze by the Coulomb potential V(r)=−4πϵ0rZe2, with no other interaction present. (a) Show that [H^,L^]=0, and explain why this forces every energy eigenspace of H^ to decompose into a direct sum of (2l+1)-dimensional multiplets. (b) Restricting H^ to one such multiplet, obtain the equation satisfied by the reduced radial function u(r)=rR(r), using the symmetric radial momentum p^r=−iℏ(∂r+1/r) (not the naive −iℏ∂r), and give Veff(r) explicitly for this potential. (c) Derive, rather than assume, the boundary condition on u(r) at r=0, treating l=0 separately from l≥1.