To finalise the development of the hyperfine structure of hydrogen in the resurrected theory, its mathematical representation is required. However, there is no need to perform any further mathematical analysis to derive the energy levels of the hyperfine orbitals, because the necessary mathematical representation is already contained within the final relationship for electron orbital energy developed in [2], {[2], Eq. (4.2)}. For convenience that relationship is repeated here.

The Lamb Shift related terms, f(G_e) and f(r_p), do not figure in the hyperfine spectra and therefore have not been expanded out here.

The ground state orbital proper is given exactly by (2.7), {with n = n_j = 1 and _en_sp = _pn_sp= n_f ^* = + 1/ 2 , and specifically designated _e E_or ( ₊⁺ )(1s)}.
Of particular interest is the relationship obtained for the 1s_h2(-) orbital in which n = n_j = 1, _pn_sp = + 1/ 2 , _en_sp = - 1/ 2 and n_f ^* is accordingly increased by unity. This relationship, in general terms is

Subtracting (2.7) from (2.8) gives the energy difference in an electron transition between these two orbitals thus

Because this energy difference is so small, it can be simplified by assuming m_e\ m₀ @ 1, so that (2.9) becomes, (m₀ is the reduced mass of the electron).

Now insert the value for | _e n_sp | and _pn_sp = + 1/ 2 to give

To obtain the final relationship, insert

Which then gives

This is the energy in the ground state hyperfine transition {1s_h2(-) Þ 1s(+)} that produces the 21.1cm emission line. To compare this with the quantum mechanics version in [4] and [5], insert [3], Eq. (3.8) for R_hy to give

and from this it is clear that d_p » 3, which clearly meets the requirement as stated earlier in Section 2.1.2. The precise value of d_p to give the exact 21.1cm line wavelength is 3.3548035.

Eq.(2.7) is used in Appendix A, together with all the results of this Section, to generate all the hyperfine, and normal, orbital energy levels for orbit shells 1 to 3 and thereby all of the normal and hyperfine spectra via allowed transitions as previously determined.

It should be noted that the analysis above does not include allowance for the air refractive index effect. This is however, incorporated in the numerical calculations in Appendix A. Finally, note that the contents of this Section answers question (v) in Section 2.1.2 above.