In both theories, the primary quantum number, n, is a measure of the single valuedness of the electron matter wave around its orbit. In this respect the two theories are identical encompassing the same tenet. To some extent the same comment applies to the spin quantum numbers, s in quantum mechanics, (QM) theory; _# n_sp in the Bohr/Sommerfeld, (B/S) theory. Both have the same numerical value but, in view of the different treatment of the electron, there the similarity ends. In B/S theory _en_sp results from the angular momentum of a real physical particle. This then leads directly to a properly defined quantised spin magnetic moment. In QM theory the spin quantum number, s, is still purported to be due to electron ßpin", but because of the nature of the electron in this theory, electron spin cannot be adequately defined. Accordingly, in QM theory the ßpin" magnetic moment of the electron is subject to the same difficulties.

The main difference between the two theories however, concerns the azimuth quantum number, (l in QM, n_f ^* in B/S). This comparison needs to be conducted in several parts because of the peculiarity in the interpretation of l.

(i) s Orbitals :- l = 0 : nf ^* = 1, (B/S Theory Spin-Orbit Coupling Omitted). In these orbitals in QM theory, azimuthal angular momentum is zero, so that accordingly spin-orbit coupling is also zero. Therefore, l = 0, and j = 1/ 2, and the orbital energy is given by, (i) the ubiquitous "zero point energy" of quantum mechanics, the " 1/ 2 " in the denominator of (5.4), and, (ii) the appearance of the Darwin term which only applies to s orbitals. The fact that there is no azimuthal angular momentum in QM theory is considered somewhat of an anomaly. Even with the electron treated as an orbiting probabilistic matter wave, this matter wave, as shown in [2], is a representation of kinetic energy, and accordingly will possess an equivalent mass in line with Einstein's renowned energy/mass relationship. The electron should still therefore generate an effective angular momentum. Consequently, spin-orbit magnetic coupling should also appear in QM theory in s orbitals.

In the resurrected B/S theory, in s orbitals, (with spin-orbit coupling omitted), n_f ^* is effectively equivalent to the zero point energy plus the Darwin terms of QM theory. It cannot be actually so equated because it represents a different concept in that it is directly related to the angular momentum of a real physical electron particle. As a result spin-orbit coupling is present as is evident in Table 3.1. This leads directly to the next Sub-Section.

(ii) s Orbitals :- l = 0 : n_j = 1, (B/S Theory Spin-Orbit Coupling Incorporated). In this case the comments in (i) concerning the azimuth quantum number l of QM theory are still fully applicable. In discussing the resurrected B/S theory however, it is now n_j that is equivalent to the zero point energy and Darwin terms of QM theory, and as is clear from Table 3.1, n_j is composed of the azimuth quantum number n_f ^* , ( 1/ 2 in s orbitals), plus the spin-orbit magnetic coupled quantum number, _en_sp, (+ 1/ 2 in s orbitals).

(iii) All Other Orbitals :- l ¹ 0, j = l ± 1/ 2 : n_f ^* > 1, n_j = n_f ^* + _en_sp. In these orbitals both theories contain exactly the same components to make up the orbital energy and the differences are related to the meaning attributed to the azimuth quantum number l. Here l is the azimuth quantum number related to the angular momentum of the electron, (in wavefunction form), thus enabling spin-orbit coupling, so resulting in the inner quantum number j. The zero point energy term is still a separate addition. On the other hand n_f ^* is now equivalent to l, this equivalence including the zero point energy term. The comparison between j and n_j is consequently identical to that between l and n_f ^* .

(iv) Other Differences. Apart from the quantum number nomenclature and meanings, and the presence of spin-orbit coupling in s orbitals in the B/S theory, as discussed above, there are three other differences to the modern QM theory. Although at this juncture they have insignificant impact on the emission spectra, they will do so when the Lamb Shift and hyperfine structure are incorporated. These differences are exemplified in the derivations in Section 3.3 and discussed there. They are:-

(a) The coupling energy between electron spin and electron orbit is only coupled into the electron orbit, not that of the proton nucleus. Hence the appearance of the term m_e/m₀ in (3.34). This correcting term is absent in the QM theory equivalent expression as represented by (5.7).

(b) The spin-orbit energy coupled into the electron orbit also, in the B/S theory, incorporates a term to reflect that effectively resulting from the proton dipole. Hence the appearance of the term
(1-g_p m_e /m_p ) in (3.34). This correcting term is absent in the QM equivalent expression as represented by (5.7).

(c) The magnetic dipole energy coupled into the electron orbit, in the B/S theory, also incorporates a term to reflect that resulting from the proton spin dipole, as represented by (3.38). No such term exists in the QM theory in the representative expression, (5.7), although this couple does receive detailed attention in the literature, [4], in discussion of the hyperfine structure for l > 0.

All of these other differences have very little impact on the orbital energy levels of the electron at this stage, but should be included because as the development continues to incorporate appropriate terms for the Lamb Shift and the hyperfine structure, they become to some extent the dominant features.