PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems · Approximation methods
The Structure of Quantum Mechanics, Ch Hydrogen, and the symmetry that explains it
Problem
Doubly-ionised lithium, Li2+, has a single electron bound to a nucleus of charge Ze with Z=3 by the Coulomb potential V(r)=−4πϵ0rZe2. The SO(4) result derived for hydrogen holds for any such one-electron, one-over-r system once Z is inserted. Consider the n=4 level.
((a)) State the energy E4 (in eV, to four significant figures) from the Casimir relation, and give the total number of degenerate spatial states and the allowed orbital quantum numbers l with their degeneracies 2l+1.
((b)) A weak uniform electric field E=Ez^ is switched on, adding H^′=eEz^ to the Hamiltonian. This operator commutes with L^z but is odd under the parity operator Π^. For each value of m from −3 to 3, state how many of the 16 degenerate n=4 states carry that value of m; this is the size of the subspace that must be diagonalised to find the energy shifts linear in E for that m. State which value(s) of ∣m∣ give a one-dimensional subspace, and explain why no state in such a subspace can show a shift linear in E. Do not compute the size of any shift.