The Bohr-Sommerfeld model has four documented failures:
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Origins of quantum theory · Symmetry, angular momentum and spin
Reading the Spectrum, Ch The nuclear atom and the Bohr-Sommerfeld model
Problem
The Bohr-Sommerfeld model has four documented failures: (i) it gives no account of how bright a line is or which transitions actually appear; (ii) a circular ground-state orbit is forced to carry L=ℏ of angular momentum; (iii) applied to helium, it gives an ionisation energy of 28.9eV against the measured 24.59eV; (iv) it has no way to produce the four and six components sodium's yellow lines split into in a magnetic field. Someone proposes a single patch: add the rule, as a new postulate, that "a jump can only occur if nϕ changes by exactly ±1". Because this rule (used by Sommerfeld to explain which lines of an atom's spectrum are seen) deals with failure (i), the argument runs: "with this one addition, all four failures of the model are removed, and the Bohr-Sommerfeld picture, suitably patched, is a correct theory of the atom."
((a)) Failure (i) has two halves, which transitions appear and how bright each line is. Say which half, if either, the proposed rule addresses. Then, for each of failures (ii), (iii) and (iv), state whether the rule addresses it, giving a reason.
((b)) Say whether the overall claim is justified, and identify which one of the four failures cannot be removed by any added rule while the electron is still pictured as following a definite orbit.