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I'd like to know if there have been attempts in solving the full problem of the dynamics of a classical hydrogen atom.

Taking into account Newton equations for the electron and the proton and Maxwell equations for the electromagnetic field produced by these charges one obtains a higly non-linear set of coupled equations. In such a nonlinear system could some feedback effects between proton and electron take place so to make possible a stable dynamics (or at least a dynamics unstable on such long time scales longer that we can consider hydrogen to be stable)? In this way the system's stability already obtained through quantum mechanics could be reproduced by a full classical approach!

P.s.: Please, as I know of the great successes that quantum theory has had since its birth, try not to answer the question telling how quantum mechanics wonderfully solves the problem.

P.p.s.: I'm also aware of the fact that electron should lose energy and that this should cause it to fall on the proton in a very short time, so please try to avoid also this argument. I asked this question to understand if the oversymplifing hypothesis', which are fundamental in solving this problem (neglect proton's motion and, as a consequence, magnetic effects) and are quiet ubiquitous in physics, wouldn't mask the potential richness that could arise from mathematical complexity.

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I vaguely remember some efforts by André Julg (not really followed on by others) in considering a non-isolated H atom in some stochastic field allegedly produced by the remaining atoms in the Universe, that could provoke the desired stability and mimic the Heisenberg uncertainty relation for this case. –  perplexity Oct 29 '12 at 10:38
    
Also, I already answered your question completely--- there is no dimensional parameter to allow a stable orbit--- you can't make a size using electromagnetic constants, the charge of the electron, and the mass of the electron/proton. So assuming a stable orbit, you get a whole family which are rescalings of each other. –  Ron Maimon Oct 30 '12 at 5:15
    
derbucher, please stop creating new accounts and posting inappropriate answers with them. You should reuse your existing account and respond using comments. –  David Z Nov 2 '12 at 16:38
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2 Answers

No, it is not possible, and the argument is simple--- there is no dimensional parameter with unit of length, so if there were a stable equilibrium at one radius, there would be many such equilibria obtained by rescaling the original solution to a one-parameter family of solutions.

In fact, it is easier to see that the stable solution is for the electron to fall into the nucleus, just from thermodynamic considerations--- the state space for the electromagnetic field is infinite dimensional, so the most likely configuration is the electron sitting on top of the nucleus with infinite negative energy, and infintely much radiated energy in ultraviolet modes of the field.

In order to get a stable equilibrium, you need a constant with dimensions length, and using $\hbar$, you get such a constant. It is not possible to do this without introducing something like $\hbar$, just from dimensional analysis, independent of the quantum formalism.

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@ Ron Maimon: I really can't see the nexus between stability and dimensional parameters. Isn't stability (or instability) a feature of the dynamics independent of the dimensional scale, provided that I preserve proportions among dimensional parameters? I mean I could put equations of motion in adimensional form and rescale them ad libitum, nonetheless if they admit limit cycles or single equilibrium points these will continue to be present irrespective from the scale I chose. Second, I haven't understood the point on thermodynamic considerations, could you give me some reference about? –  derbucher Oct 29 '12 at 9:31
    
I don't know what "nexus" means, but I can explain--- you can't have stability, because the stable orbit has a certain size. What is that size? It must be made from the constants involved, the charge of the electron and the mass of the electron. You can't make a length out of those. So if you have one solution, you have another at a different size. QED. The thermodynamic considerations are the "ultraviolet catastrophe", googling this should bring up thousands of books. –  Ron Maimon Oct 29 '12 at 12:19
    
@ Ron Maimon: "nexus" means link. I don't think there is a link between the pure dynamics of a system and system of units you choose for describing the problem. It's obvious that orbits have a size but you could express such a size by the dimensional constants you prefer, and this won't affect the very existence of the orbits, in other words I think you are wrong about the fact that you must introduce ℏ to make stable orbits pop out: if the equations of motion predict the existence of stable orbits, these orbits exist whether I write their expressions in terms of ℏ or of every other dimensiona –  derbucher Oct 29 '12 at 14:23
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@derbucher: Ok--- then you don't understand dimensional scaling. It is impossible for an equation with a scale symmetry, like electrodynamics, to have a solution with a given size. It can at best have a whole family of solutions whose nexus with each other is that they are rescalings of each other. This means the equations cannot predict the existence of stable orbits of a given size, it can only predict the existence of semi-stable orbits at best (but it doesn't even do that by the ultraviolet catastrophe argument). –  Ron Maimon Oct 29 '12 at 16:40
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It is not an "oversimplifying hypothesis" that allows us to neglect the proton's motion, it is simply a choice of reference frame. Since electromagnetism is a relativistic theory, this choice can have no effect on the physical predictions. How could the electron fall into the proton in one frame but stably orbit it in another?

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