The aim of this page and of the followings is to show the interest to take
account of the phase wave in physical demonstrations, as much for avoiding postulates as for simplifying calculations,
or more for its explanatory power. For that we will take again already made demonstrations and will present new ones
Definition of the phase wave
The phase wave idea was brought in by Louis de Broglie in his 1924 thesis but,
like many obviousnesses in physics, it was haughtily ignored, doubtless because it introduced the possibility of a velocity higher than that of light. In the framework of our model, the phase wave appears when the particle is in motion. It is due to a shift in the superposition of the two basic waves, the inwave and the outwave.
Let's see that : while in motion at a speed v, an observer at the centre of the set of waves feels forwards and backwards a DopplerFizeau effect, but this effect differs depending on whether the wave is received (inwave) or is emitted (outwave).
For the inwaves and our observer looking forwards , he is in the case of moving toward the source, even if this one is external, extended and remote. The frequencies are increased proportionally to the sum of the two speeds (c+v)/c. Looking backwards now, they are lessened according to the difference (cv)/c. At the same time, the frequencies of the outwaves are also increased forwards, but according to c/(cv), it is now the source that moves. And backwards , the frequencies are lessened by c/(c+v). The two waves, received and emitted, are modified differently, and that gives rise to a beat wave or phase wave.
Calculation shows this phase wave travels at speed V such as :
V.v = c^{2}
v being the speed of the particle and c that of light.
This formula displays obviously that when v is
small, the particle being at rest or in slow motion, V tends to infinity, and when v tends to light speed by inferior values, V tends to this same limit by superior values. The frequency of the phase wave taking a mean value between those of the two basic waves, its wavelength Λ varies like the velocity. When the particle is at rest, Λ is infinite and can stretch to the whole Universe ; when the speed of the particle is closed to that of light, Λ becomes very small.
(See an animation done with MATH CAD software).
We see at the top in green the convergent inwave, at the bottom in blue the divergent outwave ; both travel at speed c.
At the centre, in red, the phase wave which is the sum of them. The small point figures the centre of the particle
that displaces very slowly with regard to the phase wave. In this precise case, the shift between the two fundamental
waves is very small and the wavelength of the phase wave is very large, the beats run very fast, at a speed well
over that of light. On the contrary, when the particle is travelling fast, the phase wave goes at a slower pace.
At speed c, both walk together.
Beats travel in the same direction as the centre of the particle, but at a higher speed V; the two waves coincide at each beat, but shift each time of one wavelength. This phase wave comes from the back and the past and catches up
with the particle; it projects itself towards the front. It holds in itself the future position of the particle before
reaching it with its maximum beat and before the particle itself actually reaches it. It is worth mentioning
that beats reaching their maximum coincides with the centre of the particle also reaching a maximum. It then appears
a sort of emulation between the particle and the phase wave that keeps the motion going. We think that this
is precisely where the conservation of momentum originates from.
We will see now with five examples the interest of the phase wave in Physics : to allow a good explanation of the
phenomena while having coinciding the description with quantitative formulation :
the stability of Hydrogen atom,
the angular kinetic momentum and the spin,
the emission of electromagnetic waves by atoms (still to come),
the calculation of kinetic inertia,
the interpreting of this fundamental physical entity, that is momentum.
And so we will shed light on the nature and on the role or the phase wave. The fact that we find
it again at the heart of great questions where, by a shifting of point of view, it bring one sudden clearing up, proves
its interest and removes any doubt about its physical reality. Furthermore, that V could be upper than c should not stop us : we are here facing such an obviousness by convergence of evidences that we can't any more let us blind with any dogma, even if it is that of the speed limit of the light.
1. Stability condition of Hydrogen atom.
The explanation of Hydrogen atom stability was given by Bohr which leant on a postulate :
The Angular Momentum m_{0}v.r of the electron in its orbit around the proton, can assume only multiple values n of the rationalized Planck’s constant
ћ = h/2π.
The demonstration we are going to do has already been led by Louis de Broglie. We take it back putting the phase wave at the heart of
the stability of hydrogen atom : it goes with the motion of the electron in its orbit and, at every lap, stays in phase
with itself becoming a stationary wave. The length of the orbit of radius r must contain a whole number n of time
the travelled path (VT) during a period T, so :
2π.r = n.V.T,
but in a stationary state, T = T_{0} = 1/ν_{0}, and V = c^{2}/v, then :
2π.r = n.c^{2}/vν_{0}.
And, since, always following Louis de Broglie, h ν = m_{0}c^{2},
m_{0} being the electron mass at rest :
2π.r = n.h/m_{0}v
and finally,
r = n.ћ/m_{0}v
This is right the Bohr’s postulate
which here is explained by the explicit introduction of phase wave, as an obvious stability condition for
the electron in its orbit. Guided by its phase wave, the electron finds itself in the same situation as if it was in linear
uniform motion, it does not radiate energy.
2. The orbital angular momentum.
We have just given the length of the electronic orbit of an hydrogen atom :
ћ = mvr/n,
showing that it is stabilized by the phase wave which goes, at speed V, with the motion of the electron, itself at speed v, according to the formula :
vV = c^{2}. Now, the orbital angular momentum J of the electron is defined :
J = mvr.
That allows us to write :
J = n.ћ
The phase wave becomes steady by taking again at each turn the same phase. Thereby, the angular momentum is an integer multiple n of ћ = h/2π. It is equal to the action unit, multiplied by the integer number n of times that the electron is repositioned during an orbit. There are as many action units as repositionings.
3. The spin angular momentum.
It is the same for the spin. La particle turns on itself at the speed of the phase wave, well higher than that of the light; that disturbs conventional physicists. If the particle is defined as a fermion, with n = 1, the phase wave makes a repositioning in one orbit (2π) ; one says then that the spin is half integer. If the particle is a boson, the phase wave makes a repositioning every half orbit (π), then 2 during one orbit. The spin is half integer. So, using simply the phase wave, we understand the values and the quantification of the spin, without any kind of mystery.
4. The momentum.
For a particle, the momentum p is :
p = mv.
Faithful to our walk on the steps of Louis de Broglie, we bring in vV = c^{2}, V being the velocity of the phase wave and v that of the particle :
p = mc^{2}/V
Then VT = Λ, T and Λ being the period and the wavelength of the phase wave,
p = mc^{2}T/Λ
and the fundamental formula : mc^{2} = hν,
p = hνT/Λ
and finally with ν = 1/T,
p = h/Λ.
It is important to see in this well known formula that Λ is the wavelength of the phase wave. As we have seen upper, it is infinite when the particle is at rest, and small when it is at large speed close to that of light. The momentum p becomes the ratio between the quanta of action for us the action of a repositioning and the length of the phase wave. So the notion of momentum becomes clear ; at every repositioning, the energy is distributed on all the length of the phase wave, as if transmitted by it. That is more explicit that the simple formula p = mv, where we don't see clearly what is the role of the mass m of the particle.
5. Conclusions
1 The phase wave is faster than light; that become a physical obviousness we must accept. We see with our model how that can be.
2 n, called first quantum number, stands for the number of time the phase wave
overlap in one loop. It is a physical phenomenon totally understandable which stabilizes electron. In
another work
I show how two other quantum numbers, p and q, fix the apexes and the focuses of interference hyperbolic fringes and
so define electronic orbitals. That confirms :
that matter is fundamentally undulatory;
that this wave nature organizes displacements and relations between particles;
and that quantum numbers intervene by imposing to relative distances to be multiples of wavelengths or of
groups of wavelengths. We fully understand that interacting vibratory systems arrange themselves mutually in order
to respect the phases. Thus, their centres are separated by whole multiples of wavelength; from what these numbers which are no longer mysterious.
Physics must accept these obviousnesses, even if they seem contrary to its dogma. It
must build itself with what observation of nature teaches it.
