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On A Free Fall Model For Bohr Orbits Using Kepler Orbits And Nature of Atoms Radiation

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I did, last year, come to derive a set of equation based from intuition, I later was informed that a similar model was pursued outside of my own model in the past, but I believe regardless, the reasoning I set my model on the exact derivations leading to those assumptions are in themselves unique. Unique as it is, it is like a starting off point to a more simpler model than the one that preceded mine. First of all, why did I pursue a free falling model of an electron the in the stable orbits of an atom?

1. Electrons cannot be at rest in a stable equilibrium

2. Electrons cannot be in motion pursuing ellipitical orbits

These two basic premises of quantum theory are at odds with each other. One thing was certain, if electrons had motion then Bohr was right:

3. Only non-stable orbits with motion can allow atoms to radiate.

So with the help of quantum theory, Bohr created a model of the atom where electrons followed ''special orbits'' and these ''special orbits'' where the ones where electrons had been able to whizz around without giving off the usual radiation we would expect from a moving charge. Later, the planetary model was superseded by wave mechanics, the idea was simple enough

4. That electrons did not move around the nucleus, instead they existed as a wave spread out statistically in space.

But with this, there was a catch. DeBroglie, the true inventor of the wave particle duality model, for all states of matter, never said that his wave mechanics specifically said this. From experiments, like the photoelectric effect and gamma scattering, we knew the particle had to exist both sometimes as a particle, other times a wave. The inseparability of the wave from the particle, lead to his famous wave hypothesis, stating that the particle was accompanied by wave. The wave itself was unobservable however, only today using very special techniques, computers and special equipment have been able to indirectly see the wave nature inside of particles. It still doesn't tell us at what time the electron would act as a particle, unless directly observed, by which time the wave would collapse and all that would remain, would be the particle. Yeah, quantum theory was weird. 

In spirit of deBroglie, I'd like to carry on his strong assertion that particles where guided by waves, so that we can  in some way rationalise the weird nature of quantum mechanics into a regime that is more acceptable for a willing and rational mind. Certainly, why cannot a particle be guided by its wave? Matter was guided by curvature in space, and it was this correspondence of the two ideas where I linked perhaps a unity between the strange wave mechanics of deBroglie to that of GR. One stated that matter told space how to curve and space told matter how to move, whereas particles told waves how to spread, and the waves told particles how to move. Maybe wave mechanics and curvature where closely related. This was my first motivation. We'll learn as my theory progresses, that the orbits described by the moving particle also contain their own curvatures, their own geometries. By inviting the weak equivalence principle, we would further learn how to allow a particle to move in an orbit without giving off any radiation. 

My derivation was at best, rudimentary. To do it, I just needed to know some basic laws, like Newtons laws, Keplers laws and some electrostatic equations of motion. 

A quick set of equations that feature in my paper which requires little format goes like this; we wish to derive the motion of the electron around the nucleus and see if it obeys Kepler's third law. We start off by writing down the general force equation in electromagnetic theory

Equations removed. Will be rewriting them.

Where [math]Z[/math] is the nuclear charge, [math]e[/math] the electric charge and [math]k_B = B[/math] as the Boltzmann constant. For Hydrogen atoms, [math]Z=1[/math]. We notice the nuclear charge cancels and by rearranging we obtain

Equations removed. Will be rewriting them.

And viola! Its that simple, we retrieved Kepler's third law for the planetary motion in which it's acceleration around the nucleus is simply

Equations removed. Will be rewriting them.

where the mass has been set [math]m = 1[/math] in the expression [math]k_B \frac{e^2}{r^2}[/math] which is the usual Coloumb force law, and [math]\frac{v^2}{r}[/math] is the expression which talks about circular motions. 

In the more complicated arguments I explain that this acceleration disappears in the ground state, again due to it being in a state of free fall... Or does it? You might come to learn that accelerated charged objects experience only radiation, and this is true. Sometimes you might hear, ''the acceleration from a free falling body,'' and so both those statements might seem a bit at odds. What it means is that the body has to be freely falling in a gravitational field, not affected by an external source that adds to that acceleration in some way, otherwise it would obey the relativistic Larmor equation for circular orbits.  There is still wave mechanics in the theory from a de Broglie guiding wave model quick would make it semi classical , but I'm interested only in the case where the wave pilots the electron as curvature and waves may be saying the same thing since again.

We make note that

[math]\frac{r^3}{t^2} = \frac{Gm}{4 \pi^2}[/math]

which is the exact result Newton would have found when he was deriving Keplers results, who actually guessed his work! Kepler not once ever derived his equations that described motion of planetary bodies, it was all trial and error. And some excellent guesswork.

We also note that [math]Gm = r(s)c^2[/math] as the gravitational parameter. Pulling the remaining constant to the LHS gives

Equations removed. Will be rewriting them.

As a conclusion by cancelling out factors we get

Equations removed. Will be rewriting them.

The gravitational parameter. I really only came to derive this after being inspired by a book by Rogers where he stated that maybe particles obeyed Kepler's laws inside the atom, so I rewrote the gravitational theory in the language of electrostatics, so I was very surprised afterwards to learn that my idea of a free falling model of an electron had been speculated more thoroughly than my rudimentary model gave. I had no insight of a free falling model before this as I drew on a theoretical model from weak equivalence to explain why the ground state electron did not radiate. Though I claim my model is quite a bit more simpler, I think it gives a clearer insight how I fell upon these ideas which appeared to exist in literature outside of my independent model;

Michał Gryziński - Wikipedia

In his work, and no I haven't read it actual paper, but there is a Langrangian derived. It appears as a Kinetic term and a potential term and the very last term describing the spin and orbit of the equation, the spin orbit equation is something I am pretty well read up on. I haven't gone as far to describe any of the results I came to in terms of a Langrangian, all that was important to me where the two equations:

Equations removed. Will be rewriting them.

Where the first one explained how the charge obeyed a Kepler orbit inside of the atom, and where the second described how the acceleration was obeyed by the Coulomb law relationship. In regards To Michals approach, I had already been postulating on how different orbits could be more accurately described and more recently suggested a correction term to the spin orbit equation as the eccentricity of the electrons orbit, a dimensionless parameter, which if an electron was moving around the nucleus, in a real way, eccentricity would become part and parcel of the dynamics


How he goes on to derive his exact Langrangian is uncertain to me (but I can make some good guesses), but the spin orbit equation already has a good write up here, https://en.wikipedia.org/wiki/Spin–orbit_interaction


Edited by Dubbelosix
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Notice also, it was easy to derive this because we draw in the strong symmetries of the forces that are encoded in each theory

[math]F = G \frac{m^2}{r^2}[/math]

Notice the appearance of mass acts like the charge in the theory

[math]F = k_B \frac{e^2}{r^2}[/math]

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Posted (edited)

I just came across this article and feels valuable to my own investigations:

https://www.grputland.com/2013/12/self-contained-derivation-of-keplers-laws-from-newtons-laws.html#:~:text=Kepler's laws can be derived from Newton's laws,they may be regarded as point-masses (i.e. particles).

Some of its derivations are a bit messy and complicated. But we get the gist. It seems to derive a very similar eccentricity equation for standard orbits. 

Edited by Dubbelosix
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