The formula, known as Balmer-Rydberg formula, expresses the wavelengths of radiations
emitted by Hydrogen atom (Hydrogen line spectrum). It was totally empirically uncovered by a Swiss teacher of mathematics, Balmer, in 1885, long before the
knowledge of electron or of proton, therefore that of atom structure, and also long before the birth of Quantum Physics in the
mid 1920s. The formula was generalized and the constant R precised by Rydberg :

where λ is the wavelength of emitted radiation, p and q,
two integers which are quantum numbers.

Guided by the idea that atom components, electrons and protons, could be
vibratory systems and therefore could produce between them hyperbolic interference fringes, I tried to establish the general
equation of them in the hydrogen atom. It appeared to me identical to this formula of Balmer-Rydberg. . Effectively if we consider only the general aspect of this equation, we can write it so:
- disregarding λ, which is the emitted radiation when electron shifts from its orbit, since we consider only stationary states of atom,
- and letting the constant R out of this demonstration; it is the constant that gives the hydrogen atom dimensions.

we have the right equation of a family of hyperbolas, defined by two quantum
numbers p and q. In order to graphically illustrate that, we can draw those hyperbolas varying p or q.

The quantum number, q. Figure 1 shows hyperbolas that have a similar apex, but with shifted focuses: p always equals 1,
while q ranges from 2 to 5.
It’s therefore a spectrum lines, Lyman’s in this very instance. Hence, the definition of a sequence of lines: it is
produced within a family of hyperboles all having the same apex but staggered focuses onto which both the electron
and the proton take a temporary position, thus defining the atom’s level of energy.

Figure 1

Quantum number p On this graphic (figure 2), we have drawn different sequences with p value ranging from 1 to 4 (Lyman, Balmer, Paschen
and Brackett); we observe now that apexes shift. In order to make things more apparent, we gave q only one value, p+1. The other hyperboles with a same apex
(or a same sequence) would appear should we give q higher integer values. In order to go from one sequence to another,
hyperboles apexes shift when p increases by an integer value, and the sequence creates itself when q varies
and focuses shift at their turn as in figure 1.
And finally, referring to classical theory, q can be regarded as the principal quantum number, and p
as the azimuthal or orbital quantum number.

Figure 2

Conclusion

In this article, following the idea that particles are vibratory systems with
a certain common frequency, we assume that
when two such particles are brought together, as proton and electron in hydrogen atom, their vibrations make up in
hyperbolic interference fringes. Balmer-Rydberg formula appeared to us as the general equation of these
hyperbolas, with two parameters p and q, which define
the position of apexes and focuses, hence, their eccentricity.

Far from being magic or mysterious, these quantum numbers are simply the very parameters of one equation that provides the physical explanation of the hydrogen atom stability and of the electromagnetic emission.

Making them varying like quantum numbers of classical theory,
we found again known frequencies of emitted radiations by electron when jumping from one orbital to another.
Grounding our work from the beginning on physical explanation, we simply introduced the calculus of emitted
radiation by
an orbital jump; we can deduce the explanation of the phenomena: the vibratory nature of particles being confirmed, the
orbits are quantified and stabilized by interferences and the emitted radiation is directly produced by the deformation
and the displacement of hyperbolic fringes following the electron orbital shift.