All of the results that have been derived here in Section 3 for the quantised energy levels of the bound electron, are well known and well documented throughout the literature. The significant difference with the derivations here is that they have all been achieved from first principles without the need for unsupported hypotheses or assumptions. This cannot be said for the original old quantum theory of Niels Bohr et al or the modern theory of atomic structure based upon quantum mechanics. Having said this it will of course be necessary to maintain this degree of rigour as the development of the resurrected Bohr/Sommerfeld theory is continued in order to maintain credibility.

The first principles referred to above are Planck's quantum hypothesis of energy and de Broglie's matter wave quantum hypothesis of momentum. Both of these have been verified experimentally. In [1] they were used as the starting point for the investigation of the characteristics of matter waves in the Relativistic Space-Time Domain D₀. The results of that investigation lead directly into the resurrection of the corpuscular theory of atomic structure pursued here. The primary results achieved, which have led to the derivation of the quantised orbital energy levels in Section 3 are reviewed as follows.

First and most important is the establishment of the link between the main quantisation criteria, the single valuedness of the orbital electron's matter wave wavelength, and the necessity that in a stable orbit, the bound energy of the electron must be constant over a complete orbit. This link provides the criteria with a necessary and sufficient formalism to fully justify its use within the overall development. Also the manner in which this link has been established has allowed visibility of the variability of the orbital electron's matter wave wavelength within a stable elliptic orbit, and how this becomes constant throughout a circular orbit. It has also provided a formal proof for the exclusion of the pendulum orbits as a result of their contravention of the primary criterion of existence within D₀.

The next most important aspect is the ease with which the development led to the relationship between the primary quantisation criteria and the physical orbital velocity of the electron. It is in fact a relativistic version of de Broglie's original quantum momentum hypothesis. It is this relationship which is then shown to be easily transformed into Bohr's quantised momentum rules although again it is the relativistically mass corrected versions that appear. This quantisation relationship is the one that is used to provide the input to the bound energy equation of the electron. It was subsequently used in this paper to provide such quantised inputs for the four orbital cases sampled in Section 3. The same relationship will be shown in the next paper to be just as applicable to further cases involving electron spin.

The orbital energy relationship into which the above quantised momentum relationships were inserted is really a partly developed Sommerfeld equation for elliptic relativistically mass corrected orbits. The version derived here does however permit the development of a complete mathematical model to be progressed in stages covering a variety of orbits as effectively demonstrated in Section 3.

It is important to note that in many of the derivations presented in this paper a number of simplifications in the form of relativistic approximations have been utilised. Most of these have been taken to the first, (v ²/c ²), relativistic term. While these approximations provide for a considerable degree of ease in the mathematical development, they will of course become the source of error, however small, in the calculation of line spectra wavelengths. The reason for the use of these approximations is essentially threefold thus, (i) mathematically necessary in that a fully rigorous solution was unattainable, (ii) ease of derivation as suggested above and, (iii) to ensure compatibility of results with those extant, i.e. Sommerfeld's energy equation. However, as further development proceeds and greater accuracy is needed, it may be necessary to refine some of the approximations used and such refinements will be introduced as needed.

Finally, everything that has been presented and discussed here has obviously ignored what is now accepted as an integral part of atomic structure, electron spin - howsoever viewed. This will be the subject of the next two papers, the first of which will deal with the mechanical aspects of electron spin, while the second deals with the magnetic. It is in the next paper that electron spin will be shown to be not just an integral part of atomic structure theory, but quite possibly the most important part.