In one of Louis de Broglie's early papers ("Sur le parallélisme entre la dynamique et du point matériel et l’optique géométrique." – J. de Physique. Série VI. 1926. 7. Р. 1.) he proves that the behaviour of what we now call "de Broglie waves" in the hydrogen-like atom (with a nucleus charge +Ze) is as if the atom behaved like a refracting sphere with the effective spherically-symmetric refractive index n(ν, r) given by the following formula:
n(ν,r) = √{(1+Ze2/hνr)2 - (mec2/hν)2}
Here me is the electron's mass, ν is the wave's frequency, r is the distance from the nucleus and h is obviously Planck's constant.
Also, very curiously, he comments in passing that such an object (which de Broglie calls "refracting sphere of Bohr's atom") manifests "the qualities of mirage/illusion".
Now, my question is: can we actually build such a sphere, so that the light rays refracted therein would behave in the same way as de Broglie's waves behave inside the atom? If we could do this, then this would serve as a very nice "toy illustration" of the wave mechanics of an atom.
The problem is: are there any known materials (crystals etc) that would have such specific dispersion law? Not necessarily with the same numerical values of the parameters, but at least having the same shape of dependence on the frequency and radial distance.
I had never heard of that before, and to be honest it is a very interesting way of expressing it. Thanks for sharing it.
If I'm reading it right, the index goes to infinity at the center. Also overall exhibits anomalous dispersion in that the index decreases with shorter wavelength.
That makes creating a sensible object sort of impossible.
This behavior might be akin to popular sci-fi visualizations of black hole.
You could probably model this in a ray tracing system. lens design system, or CG rendering system. Some of these systems would also allow you to produce an image as if you were looking through it.
Dear Tigran,
It is quite unfortunate for physics that de Broglie's many papers and notes to this date have not been translated to English.
In other papers, notes and many books of his, he clarifies that what you mention as a "refracting sphere", corresponds in reality to a "resonance sphere" that appeared to him as being located at the mean Bohr radius, and corresponding to an axial resonance motion of the electron about this mean ground state radius.
It is this idea that gave Schrödinger the idea of representing this "resonance state" with a wave function.
This is precisely this discovery by de Broglie, that was left unexplored in the community as soon as the statistical method of Heisenberg was published and readily adopted by the community as being a more precise representation, that I have been working at bringing to the fore again with my last paper, titled "The Hydrogen Atom Fundamental Resonance States", that recalls the historical context of de Broglie and Schrödinger discoveries in this regard with quotes of their specific comments:
http://file.scirp.org/pdf/JMP_2018042716061246.pdf
Best Regards
André
Thank you André Michaud I will definitely read your paper. Btw, I have read most of Louis de Broglie's books and as for his papers --- all of them have been translated into Russian and published recently (just a few years ago) and I have 3 volumes of the 4 volume set. The first volume I have finished reading already. Btw, it also contains the wonderful book "The Prince in Science" by Georges Lochak, which I enjoyed immensely, because most of his (de Broglie's) ideas of which I never suspected while being a physics student in the University, because we were never taught these things in standard courses of Quantum Mechanics, happened to coincide with my own "independent ideas" which I developed while studying the "orthodox" Quantum Mechanics. In my current research work on "quantum infodynamics" I try to realise de Broglie's ideas further, but going beyond the pure states, i.e. I postulate the fundamental description by density matrix as "a state" and that leads to a much more coherent framework, connecting with the nonlocal statistical mechanics also (in the spirit of another great physicist, this time the Russian one, namely A.A. Vlasov).
Dear Tigran,
Glad to hear that you are so into the de Broglie's writings. I didn't know that they have been translated to Russian. Glad to hear this. At least a sizable group of the most advanced physicists in the world now have access.
I think you will be in for a few surprises reading my paper, because all of my analyses are grounded on de Broglie's deep understanding of Maxwell's electromagnetism. Not to be confused with electrodynamics, which is grounded on Lorenz's views, with which Maxwell disagreed totally because Lorenz did not understand the mutual induction of both electric and magnetic aspects of energy on which Maxwell's electromagnetism is grounded, and that results in the ability to directly calculate the speed of light as the only possible velocity in a direction orthogonal to the transverse mutual induction of both electric and magnetic aspects that Maxwell has intuited could be the only electromagnetic structure that could logically explain propagation of light.
The most interesting outcome of de Broglie's understanding of Maxwell's electromagnetism is his success in defining what electromagnetic properties a localized electromagnetic quantum needs to have to remain permanently localized as it moves in vacuum. This allowed finally ironing out the last remaining hurdles and finally establishing the equations that he was looking for:
https://www.omicsonline.org/open-access/on-de-broglies-doubleparticle-photon-hypothesis-2090-0902-1000153.pdf
I also have the really fine book by Georges Lochak "Louis de Broglie - Un prince de la science". I purchased it 25 years ago shortly after it came out. It is still being published and available in French:
Lochak G. (1992) Louis de Broglie - Un prince de la science. Collection Champs. Flammarion. France. ISBN 9782080813374
I even had the privilege way back of having Georges Lochak personally answer a question I had emailed to the Foundation Louis de Broglie to get clear about an issue that I couldn't ascertain about de Broglie's conclusions regarding his localized double-particle photon hypothesis, as I was analyzing his conclusions.
The anglo-saxon community has no idea what de Broglie discovered.
Best Regards
André
It is easy today to make any optical gradient, but to make it also precisely dependent on the wavelength as required here is much more difficult. Why is here a need for wavelength dependence, what determines the spectral content of the wave bunch orbiting the hydrogen nucleus?
Mario Stipčević This dependence is dictated by the energy E= hv and de Broglie's wavelength relationship as applied to a massive particle (electron orbiting nucleus in an atom), see Section 3. of de Broglie's paper I referred to.
I see. The apparent wavelength is function of both mass (which is constant) and momentum which I guess is changing from one instant to the other and is in relationship with the distance from the center. This is quite a complicated picture, but, can we say that if we replace point-like photon with a deBroglie's wave, we have a completely classical picture of a wave bouncing around a funny sphere?
That was exactly my point (or guess) --- could we build a classical (in the sense of "photon wave" obeying classical Maxwell's electrodynamics) model of an atom by substituting de Broglie's waves with the light waves in an artificial medium with the appropriately constructed non-homogeneous refraction index? If so, that would be a "very nice little toy" to have :)
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