A particle moves in a rotationally invariant three-dimensional potential
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group
Problem
A particle moves in a rotationally invariant three-dimensional potential V(r), so that H^ commutes with every component of J^: [H^,J^x]=[H^,J^y]=[H^,J^z]=0. Let ∣E,j,m⟩ be a simultaneous eigenstate of H^, J^2 and J^z with H^∣E,j,m⟩=E∣E,j,m⟩, and let J^±=J^x±iJ^y.
((a)) Using only [H^,J^±]=0 and the fact that J^± raises or lowers m by one unit without changing j, show that J^±∣E,j,m⟩ (when non-zero) is again an eigenstate of H^ with the same eigenvalue E. Hence state how many linearly independent states with energy E a given (E,j) multiplet contains.
((b)) The one-dimensional oscillator Hamiltonian is H^osc=ℏω(N^+21) with N^=a^†a^, and the raising operator obeys [H^osc,a^†]=ℏωa^†. Explain, using this commutator and the general connection between a commuting generator and a conserved quantity, why stepping up the oscillator ladder with a^† changes the energy while stepping up an angular momentum multiplet with J^+ does not.