The Fine Structure Constant revealed.
Rutherford was the first to conceive of the hydrogen atom as a planetary model made up of a proton and an electron. The proton took the place of the Sun around which the electron orbited like a planet. The force of gravity linked the planet to the Sun and allowed it to travel along a stationary orbit while the centrifugal force kept it from falling into the Sun.
Similarly, the electron, attracted by the proton’s electrical force, did not fall onto it because of the centrifugal force that opposed the attractive force and traveled as a result on a stable orbit around the proton.
To maintain the necessary centrifugal force, the electron had to revolve around the proton at a very precise velocity. To explain the atom’s stability, its value could not be less than that very precise velocity. If it had a lesser velocity, the electron would be attracted by the nucleus’ positive charge, whereas it would leave its orbit along a tangent if its velocity of revolution were greater than the one resulting in equilibrium.
It very quickly became clear that the classical theory of electromagnetism was incompatible with the stationary quality of the electron’s orbit as an electric charge.
According to classical electrodynamics, the orbiting electron would have had to emit electromagnetic radiation as it was subjected to radial acceleration by its motion along the orbit. It would thus lose kinetic energy during is circular motion around the proton and as a result progressively lose velocity to end up falling onto the proton when the electrostatic attraction came to predominate over the centrifugal force.
To get around this difficulty, Bohr proposed to use Planck’s idea of quantizing energy. He postulated the existence of discrete quantum states of the energy of the electron that would make it revolve in stable orbits without emitting radiation or losing energy.
The acceptance of these stationary states, which were originally based on the mechanics of Heisenberg’s matrices, was not received well in the scientific world.
The physical interpretation of the matrices was obscure and the majority of physicists at the time considered it somewhat distasteful for physical and mathematical reasons.
De Broglie’s thoughts about waves and Schrödinger’s famous wave equations explained better the fact that the energy levels remained stationary .
De Broglie’s wave hypothesis, which accepted that electrons in motion could possess a wave-like nature and be described accordingly, appeared to be more acceptable because it was more familiar mathematically to physicists.
And above all it presented an image for the physical interpretation of the atom based on waves that could be received more immediately as a model and thus was more satisfactory intuitively.
But already Schrödinger’s contemporaries were aware of the ambiguous nature of the interpretation of his formula.
In 1928, Arthur S. Eddington wrote:
Schrödinger’s theory is enjoying great popularity now, in part for its intrinsic merits, but in part also, in my opinion, because it is the only one of the theories (that exist right now) that is simple enough to be misunderstood.
Eddington had understood fully the ambiguity and the danger implicit in the Copenhagen School’s intentions .
The decisive battle in fact played out over the physical interpretation of Schrödinger’s wave formulas that formalized de Broglie’s description using waves.
The interpretation of matter at the elementary level was in doubt because of the ambiguity between the two possible interpretations of Schrödinger’s formulas.
In the conflict that followed the presentation of his Wave Mechanics, Schrödinger’s equations were interpreted as being functions of “probabilities” rather than “real wave functions”.
Leaving to one side the unresolved problem of the Fine Structure Constant, Quantum Mechanics initiated its series of probabilistic interpretations of the atom’s quantum system.
It became altogether impossible to represent the orbital model of the atom using the wave interpretation, which was finally downgraded to no more than an ideal mathematical representation of the probabilistic interpretation imposed by Quantum Mechanics.
Quantum Mechanics’ overwhelming force gradually eliminated all opposition from the scene and today opposes every new theory strenuously and excludes it from the models it has by now institutionalized.
We now wish to try a new approach to the interpretation
of the atom’s quantum levels.
As the Wave Theory of the Field opens up new horizons to us, we have a new opportunity to try to understand the physical reasons which make the value of the fine structure constant what it is and not something else. The new theory allows us to indicate which of the atom’s truly physical wave properties determine it.
The novelty of our approach is that the wave model of the electron introduces a new, strictly physical vision of the electron’s orbits in Bohr’s atom, which provides us with a strictly causal explanation of the radiation emission.
The structure of the atom in terms of waves provides us with a new explanation of atomic emissions and in the end gives us an easy and unambiguous understanding of the reasons behind the specific value of the fine structure constant.
Let us see how this is.
Think of an electron in motion in space. It can be described as a wave train moving along a cylinder whose axis coincides with the direction of the electron’s motion. Let us describe it initially in terms of the non-relativistic wavelength of the waves with wavelength le that precede it along its trajectory in the cylinder centre.
To understand the electron’s role within the atom more easily in terms of waves, we must describe it schematically as a wave train originating from a wave source that produces two series of wave surfaces that propagate in opposite directions along the orbital cylinder:
- forward in the direction of its motion, and
- backward along the trajectory it has already traversed.
For the electron’s waves to be in a resonance state in its orbit around the proton, its wave surfaces must be perpendicular to the trajectory traversed.
- When we look for possible states of stationariness in terms of waves in the orbit of the electron, we see that these are dictated by the wave laws of resonance that require there to be a integer number of wave surfaces surrounding a closed orbit.
- Even de Broglie’s Wave Mechanics require waves associated with an electron in orbit to be in a state of resonance.
Picture 63. An electron in the orbital tube around a hydrogen atom.
- The difference between the two theories is that de Broglie’s waves only exist at low energy levels with a single wave occupying an entire orbit.
- It in fact describes only the kinetic energy of electrons in motion along the orbit and fails to describe electrons’ waves of mass.
- We will now describe physically all the elementary waves of the electron wave source that occupy the entire orbit at a higher energy level and thus with a greater number of wave surfaces.
We can also use classical mechanics to arrive at the resonance state, as Bohr did, using the wave interpretation of the Principle of Minimum Action that conditions any body’s motion through space. It is worth pausing on this subject for a while before moving on to our original discovery of the wave-like nature of the atom.
In an extremely pleasant popular science book with pronounced mystical connotations “Universal Constants in Physics” that was published recently, the authors, Gilles Cohen-Tannoudji and Michel Spiro illustrate the Standard Model of the subject and seek to link it with very little success to the relativistic concept of space-time.
To do this, they make some reference to the logic of Quantum Mechanics and many other references to the Berkeley idealism of the old school.
They have unlimited faith in the mathematical generalizations of pre-quantum physics and in the inevitability of successive developments in Quantum Mechanics. At one point, they even assure the reader that:
No, there was no alternative. Pre-quantum generalizations led physics inevitably to Quantum Mechanics with its interpretations of the dual model …
In terms of particles, the most important generalization of a Lagrange formula was the one involving electromagnetism. It is in fact possible to demonstrate that Maxwell’s equations, which are the fundamental equations in electromagnetism, derive from the Principle of Minimum Action.
We could describe Maxwell in a Lagrangian formula as a function of the electromagnetic field, which is derived as a dynamic system with infinite degrees of freedom that interposes a “kinetic energy” due to the propagation of the field.
The role played by this generalization is of capital importance, as it allows us to do away with a medium as the locus of the propagation of the electromagnetic or luminous waves.
In some way, this marks a return of the luminous phenomena that predominated prior to Maxwell’s equations and which were conceived as corpuscular: if no medium exists to support the waves, the thing that propagates must after all be material and thus possibly corpuscular.
This conclusion is self-evident to someone with a positivist spirit and clearly suggests conclusions to minds that have been conditioned properly by Quantum Mechanics.
“As the thing that propagates transmits energy, and we only know the capacities of bodies “possessing kinetic energy” to transmit energy, we must necessarily accept that any entity that transmits energy is a body.”
This is followed by their final argument:
Although Maxwell’s equations by their nature deal essentially in terms of waves, their derivation from the Principle of Minimum Action make it possible to give them a corpuscular interpretation.
Although this conclusion appears to indicate a possibility, the authors in their hearts have no doubt that the possibility of a corpuscular interpretation they suggest is already an established certainty that has been held for some time.
But they did not yet know that even the Principle of Minimum Action can be understood better by the new Wave Theory in terms of waves.
The mysterious nature of the Principle of Minimum Action has perturbed generations of physicists to the point of giving destiny itself a mysterious role in physics. Lagrange’s corpuscular interpretation seemed to suggest that nature was pursuing a goal in its mechanical actions.
The Principle of Minimum Action appears to impose on any body that moves from one point to another in space a strange economical behavior that tends to reduce to a minimum the expenditure of a parameter called “action”.
As is well known, action is the product of a momentum multiplied by a length:
S = m v r
(where the momentum is defined as mass “m” per velocity “v”, and length “r” is the space traversed and S is the action).
Among all the possible trajectories that could take it from its point of departure to its destination, a body chooses the trajectory that reduces the “integral of action” that can be calculated for each trajectory to the lowest level.
It is in effect the integral of a quantity or the difference between the body’s kinetic energy and the potential energy of any possible force fields present in the space that it traverses in its trajectory that is called “Lagrangian”.
Stripped of hypotheses positing mysterious goals, the action of the principle has remained an observation pure and simple of the way nature behaves. Scientists do no more than take note and generalize what they see in order to use it in predictions or in calculating the trajectories of bodies.
It is quite a different thing to determine causes. Causes remain inexplicable in classical, relativistic and Quantum Mechanics.
The Wave Model gives us an opportunity to get at the causes, however. In it, bodies as such are no longer limited or even punctual entities but wave sources with wave fields that expand into the surrounding space.
The Principle of Minimum Action appeared to make it possible to associate something like a goal to bodies in motion. They appeared to have a consciousness at any point of their trajectory of what would happen throughout the journey they had to make.
Bodies appeared to be entities that extended along the entire journeys they had to make that adapted their motion from beginning to end to make the journey take as little time as possible in keeping with the force fields they had to traverse.
This last apparent absurdity contains the seeds of the
essence of bodies’ wave-like reality.
- It is true!
The wave fields of bodies as wave sources do somehow know every point of the journey they are facing. The behavior and propagation of their waves that precede them at the speed of light are in fact conditioned by the geometries they encounter along their paths. - It is true!
The waves’ behavior is certainly a function of the bodies’ mass at rest. This after all determined their wavelengths at rest. - And it is just as true
that this is also a function of their velocity. Velocity after all conditions their Doppler wavelength. - The length of their journeys also matters.
These will contain a certain number of wavelengths, which, should they achieve a particular resonance state, will condition their future behavior. - They will thus be as important as the fields the waves traverse.
The waves will be deviated by the geometries these fields impose on space. And they will involve thewave mechanism that constitutes the source that produced them in this deviation.
There are thus many good reasons why
the Principle of Minimum Action
should be a wave and not a corpuscular principle.
In his analysis of electrons’ motion in the atom, Bohr began by confirming that, according to classical mechanics, a mass in motion in the space between two points behaves in such a way as to develop its trajectory as a function of the Principle of Minimum Action, which states that the product of the mass of the body multiplied by its velocity and the distance traversed should be as small as possible.
In an orbital pat that is closed on itself and S is the quantity of action, we find:
S = me ve 2π ra where ra is the radius of the orbit traversed, ve the velocity of the electron and me its mass at rest.
As we know from Planck that every action is made up of whole multiples of quanta of action “h”, the formula for the action must be S = n h divided by n = 1, 2, 3, … and thus:
n h = me ve 2 π ra
Picture 64. Planetary model of Bohr’s atom.
To interpret the nature of the electron in terms of waves, simply substitute the mass of the electron in this formula with its equivalent in waves:
me = h / λe c,
This gives us from the state of minimum action the law of resonance for the elementary waves that circulate along the first orbit of radius ra in the hydrogen atom:
2 π ra = n λe c / ve.
This formula (2 π ra) describes the length of the orbit traversed by the electron around the proton as a function of the number of wave surfaces present on the orbit.
As you can see, this is ” not ” an integer number:
c / ve = 137.03599976(50)
With an incertaity on the (50). given the ratio between velocities “c” of the waves and “ve” of the electron wave source, which is necessary for the electron itself to possess sufficient centrifugal force to resist the proton’s electrical attraction.
As there is a resonance state, this ratio must yield an integer number.
If we then substitute the velocity of the electron measured in a Rutherford equilibrium state in the last formula, we get the value of the radius of Bohr’s atom.
The physical significance we can attribute to the formula, which describes the stationary state of the orbit traversed by the electron, is essentially that: 137 wave surfaces of wavelength λe are circulating on the orbit traversed by the electron.
Picture 65. Formulas of the atom in terms of waves demonstrate that one can formulate a quantum description of the energy states occupied by the electron in the atom that is at the same time based wholly on waves.
It also tells us that for each complete revolution of the electron, the wave surfaces that precede it along the trajectory at the speed of light complete 137 revolutions.
Is that all? It seems almost too easy. This could be the banal explanation of the significance of the fine structure constant.
It seems we are actually getting close, but this is not yet the complete explanation. There is something that suggests we are on the right road to find the true nature of the fine structure constant, though.
But, the description of the electron wave source we provided is not yet relativistic.
To be consistent with the dictates of Relativity, we must complete our description in terms of waves and describe electrons’ wavelengths in terms of Relativity and the Doppler effect.
We will then assess whether this has made our model more consistent.
When we apply Relativity and the law of resonance to the electron wave source in stable motion in orbit at velocity ve ,we observe the waves that precede the electron to possess the relativistic Doppler wavelength λe1 , which is shorter than λe.
This Doppler wavelength fits 138 times in orbit “K” around Bohr’s atom.
Furthermore, when we examine the waves that propagate on the same orbit (in the same ideal cylinder) but in the opposite direction to the motion of the electron, we see that, at the same velocity of the electron wave source, the opposite relativistic Doppler effect increases the electron’s basic wavelength, which becomes λe2
This wavelength is longer than λe and fits 136 times in the same orbit of radius ra.
There are thus two wave trains propagating on the orbit of resonance in opposite directions in the orbital cylinder moving in opposite directions to one another on the same orbit.
The wave train with the shorter wavelength occupies the space with 138 wave surfaces, where as the wave train with the longer wavelength does so with 136 wave surfaces.
Let us grant that both wave trains exist in a state of resonance and parallelism on the same orbit with the electron wave source.
We can observe that as their energies differ only slightly, the two wave trains traveling in opposite directions overlay each other in the orbital cylinder producing one of the most characteristic phenomena of superimposition due to the interaction between waves: a “beat”
Picture 66. Wave of the electron in the atom on a resonance state.
One effect of the superimposition of two elementary waves with wavelengths that are only slightly different is in fact the production of a “beat”.
This is a wave phenomenon involving an increased density of elementary waves very similar (but essentially different in its interpretation) to a wave phenomenon involving interference between sinusoidal waves commonly known as “beats” in radio broadcasting.
We often observe the existence of a “beat” when we listen to a station on the radio that uses a wavelength very similar to the one used by another station transmitting at the same time.
The radio’s antenna receives the two waves together and they become superimposed on each other for periods of time producing a phenomenon called interference constructing a new wave that we perceive as a sharp whistle.
The crowding together of elementary wave surfaces on the electron’s orbit of resonance provokes a perturbation in the uniform energy level present on the orbit.
This propagates as “wave variation” along the orbital cylinder and traverses the orbit in the electron’s direction of motion at its same velocity. A new, independent wave is thus created with a modulated frequency and a wavelength equal to the length of an orbit of radius r a. This modulated wave appears to accompany the electron in its orbital motion.
Picture 67. A classical beat is not the same thing as the crowding together of two-dimensional surfaces, but they can give us an idea: in fact, we are dealing with the localized crowding together of elementary wave surfaces.
This wave variation, which is the product of a special “crowding together” of the wave surfaces present on the orbit of the atom, is similar conceptually to a beat between the two waves that superimpose themselves on one another.
But this phenomenon occurs without any effective interference
between the elementary waves and can as a whole
be identified specifically with the waves associated by de Broglie
with the electron in Bohr’s atom.
It is the same wave that Schrödinger describes as physically real in his Wave Mechanics.
Quantum Mechanics on the other hand considers it to be a mathematical function of the “probability” of finding the electron at a particular point of the atom and only incidentally identical with the mathematical description of a real wave.
The average of the wave surfaces present on the orbit (136+138) / 2 brings the average number of wave surfaces present to 137.
Everything seems to fit the same model de Broglie used in his Wave Mechanics to provide a wave interpretation of the quantum states in Bohr’s atom.
But the new model is more complete and reflects a different approach by introducing the wavelength of the elementary waves. This, among other things, allows us also to explain the hitherto inexplicable condition of double wave energy in Schrödinger’s atom.
This is in fact predicted by the presence of two waves of different energies on the same orbit, the existence of which was anything not easy to comprehend in the context of the old Wave Mechanics.
It is now easy to understand the simultaneous existence of these two waves in the resonance state that propagate in opposite directions along the orbital cylinder on the axis of electron’s path.
This also allows us to observe the “physical” conditions for the superimposition of the waves, which we had previously simply postulated as formal conditions, from which we derived de Broglie’s wave description, on the basis of the hypothesis of superimposition Claud Elbaz had proposed in purely mathematical terms.
At that time, the two waves were supposed only to exist as waves that could be superimposed “formally”. Here, on the other hand, we see the conditions under which the two wave systems with different frequencies can actually be superimposed physically and still exist simultaneously as they revolve in the same “orbital, circular cylinder” that contain the resonance orbit, in(Picture 66 )
And we can clarify yet another mystery by referring to waves, a mystery that this time lies at the very heart of the field of Quantum Mechanics.
The probabilistic interpretation Quantum Mechanics provides of Schrödinger’s wave function ψ is supposed to establish the probability of finding the electron in a particular position surrounding the proton.
Eddington expressed the doubts that arise concerning the probabilistic interpretation of the ψ function clearly as early as 1927:
This probability is stated to be proportional to ψ2 rather than ψ. The interpretation is very obscure, but it appears to depend on considering the probability after one knows what has happened or the probability for the simple purpose of predicting.
We calculate ψ 2 by introducing two symmetrical systems of waves ψ traveling in opposite directions in time. One of these must presumably correspond to a probabilistic deduction based on what is known (or is stated) to have been the condition at a subsequent time.
Until now, this mystery and doubt has never been resolved physically even in the context of Quantum Mechanics. We can no longer have any doubt about the “physical” interpretation now, however
The two waves actually exist and “really” travel in opposite directions in space-time on the atom’s resonance orbit. Even someone who is crazy enough to want to continue with the probabilistic interpretation would have to recognize that the new wave model improves our understanding of the retarded or anticipated probability function.
The explanation of the fine structure constant is no longer banal now. It is almost complete, but it is not yet completely precise. Experimental findings do not in fact give us the integer number 137 necessary to prove the ideal resonance state, but the non-integer number 137.035999(50).
Our description using waves alone no longer needs Bohr’s concept of the “complementarity” of a dual treatment, which, by definition, seeks to mix the corpuscular appearance of Bohr’s atom with its “only apparently wave-like” nature.
With the Wave Theory of the Field, we can use our description instead to find a causal interpretation of atomic emissions and thus a deterministic explanation of the photoelectric effect based strictly on waves.
To do this, we must first understand the difference between the graphic image of an electron wave source moving along a resonance orbit and the image of the same wave source moving along an ordinary “non-resonant” orbit.
Picture 69. Movement of an electron on a non-resonance orbit.
Some might question whether it is legitimate to let a simple geometrical illustration of an intellectual experiment guide us, if only qualitatively, and there are certainly some who would say it is not.
We would counter that the success we have had up to now with this method justifies it. And we should not forget that we are trying to approach physics through geometry. The only entities proposed in this theory are time and space, and the fact is that “geometry is drawn”.
The figure representing the non-resonant case is so suggestive and so full of explanatory elements that the difference between it and the graphic representation of the resonance state is obvious and unmistakable and overcomes any prior objection.
To draw the phenomenon of the “resonant wave”, the draftsman must show:
- that the velocity of the waves originating in the electron are greater than the velocity of the electron,
- that the wave surfaces are parallel to the resonance orbit.
On the other hand, the representation of the phenomenon of the electron’s movement along a “non-resonant” orbit requires only the first condition: that the velocity of the waves originating in the electron are greater than the velocity of the wave source.
The result of the non-resonant state shows that the emission of the spherical wave is the product of a “modulation of the frequency” of the electron’s elementary carrier wave. This wave expands like a spherical spiral with a form in every way equivalent to the evolving spherical plane formed by the electron’s elementary waves.
Nature is stingy. And the observance of the Occam’s dictat it is wanted.
When it finds an economical form for its structures, it does not hesitate to reuse it frequently.
This wave is spherical and is the equivalent of the emission of radiation from a dipole that is closed on itself in the form of a ring. We need to understand the mechanism that leads to the emission of a “single” linear photon, emitted like a wave made up of wave variations organized in a brief wave train. In our explanation of the photoelectric effect using only waves, we will see that the wave variation explains its actual physical existence.
Have we now really understood the fine structure constant?
Yes. We firmly believe that we have understood the periodic mechanism nature has contrived to give the atom a stable wave-like structure.
To look at it in a coldly rational light, it seems that nature had no other possible alternative. It could only have constructed the atom as we have described it given what it had available.
The same law of resonance that made possible the inevitable construction of the Spherical Involvent determines the conditions of the stationariness of the electron moving around a proton on the simplest possible orbit.
If we look back and assess incredulously the path we have already traversed beginning with the quantized structure of discrete space-time, we may find it hard to “accept” the reasons for the existence of the Ivolvent-spherical particle.
But once we have accepted that anything that can occur logically does occur, any rational organization of the structures possible in discrete space-time seems coherent and inevitable to us.
The logic we use to analyze the coherence of nature should not be considered solely and fortuitously human.
We have used Reason to construct a system of logic derived inevitably from the logic of nature, and Reason tells us that “rational” means “logical” and thus consistent with the logic of what exists.
On this basis, we can reassess “Reason” and the human method of rationalizing the structures of nature.
As we proceed along the path we have taken, we will realize that we are being forced to link each new structure to all the ones that preceded it, and these links will become increasingly difficult to follow, as the number of possible variants increases exponentially.
The variations in turn will produce an amazing variety of organizations, that will obey a increasingly complex logic of their own and create new rules of association and combination, new laws and new levels of rationality.
From now on, the simple and logical foundations of the rationality of nature will never elude us.
Whenever we entertain doubts about our “own” rationality, we have only to examine a parallel or comparison with the rationality of nature and judge accordingly.