Further Simulation Proof of Hydrogen's Non-relativistic R_H

I decided to further constrain the bounds of velocity perturbations in my
hydrogen simulation according to what I think is theoretically correct. A
basic tenet of quantum physics is that bound conserved motion can only
change in energy in integral jumps. The specific jumps that are allowed have
nothing to do with expectations as in QM, they are simply physical
constraints on how small, bound conservative systems can change energy,
based on the foundational work of Planck.

In an orbit theory, there are constants of motion, the two most important
being total energy and total angular momentum. These values do not change,
for example, in Newtonian mechanics (or Coulombic mechanics for
charged-based motion in an electrostatic field (no magnetism)). In pure
Newtonian theory, orbits are "perpetual motion machines," forever going
round and round with total energy and total angular momentum never changing.
When jumping to special relativity for orbit modeling the hydrogen atom,
Sommerfeld equated the relativistic total angular momentum to n*h_bar,
meaning what was conserved was n*h_bar*dt/dtau with dt/dtau equaling the
specially relativistic 1/sqrt(1 - (v/c)^2). Bohr at first did not do this,
equating instead the simple non-relativistic total angular momentum to
n*h_bar. Who got better answers? Even back around the early part of the 20th
century, we could see that hydrogen did not outwardly manifest relativistic
effects, so I wonder what was thought about that back then.

What I've done is to extend the orbit work of Sommerfeld. The first model
that is fully differentially geometrical is a Schwarzschild representation
of the electrostatic Coulombic field surrounding and generated by the
proton. There are closed formed solutions to the radius and velocity for
each spherical shell, just like there are these corresponding equations for
the Bohr and Sommerfeld theories. Please see the paper at
http://sb635.mystarband.net/unified/res_2.pdf for a description of
hydrogen's electronic Schwarzschild field. Also note at the end, I've
presented entirely the wrong R_H value as the observed. If any of you
reading have followed this R_H quest of mine, in my defense, it is extremely
difficult to get anyone to agree on what is an unbiased observed R_H value.
I think I have proven, though, the best accepted value is non-relativistic
in signature. And I'm pretty sure I understand why now. In the above paper,
I called the R_infinity value the observed, which is obviously incorrect.

The full differential geometry extension of the electronic field surrounding
the proton is Kerr in nature. There is a scaling of a Kerr field that works
for modeling hydrogen that I describe in the above paper. In a Kerr
geometry, an angular momentum parameter related to the angular momentum of
the central body (proton) (this is not the total angular momentum of the
electron in orbit) enters the metric and further curves the space
surrounding the proton. This is in fact this model's representation of
magnetism. In these relativistic orbit models, the correct total orbital
angular momentum to be conserved is the total relativistic orbital angular
momentum. I further programmed my hydrogen simulation to constrain the total
electronic Kerr relativistic orbital angular momentum to integral multiples
of h_bar (see eqs. (3) and (5) of my paper). This was done after a velocity
out-of-plane perturbation was imposed. First of all, the quantum jumps to
greater than n = 1 went away. I think I know why, but the interesting thing
is how various beginning radius (r) values behave. I have three I use,
Bohr's classic non-relativistic r, Dirac's r based on the corresponding
radius value of Dirac's predicted ground state binding energy (which is
identical to Sommerfeld's r; see the paper, Table 2), and a third produced
for the closed-form electronic Schwarzschild equations (19) and (20) of my
paper (Table 2).

The first simulation I ran was using Dirac's r in a Kerr field parameterized
to produce a correct hydrogen model. No out-of-plane velocity perturbations
were introduced. The simulation started the perfectly circular obit in the
x-y plane (no inclination), and stayed in plane. With no perturbations, the
simulation is essentially perfectly deterministic, and the radius stayed at
Dirac's r. There is no inherent purposeful quantizing of orbital energy or
angular momentum anywhere in the program. This is continuous physics at
first (no perturbations) and whatever is the circular initial state, the
program integrates forward in a stable manner. This was also true for
starting with Bohr's r and the r from the electronic Schwarzschild equations
(eqs (19) and (20)). A graph of one of these perfect orbits is at:

http://sb635.mystarband.net/fig1.pdf

The circular orbit (there are actually about 10 plotted, all sitting on top
of each other) looks elliptical because of perspective and axis scaling.
Next I introduced the degree of chaotic electron motion by turning on the
out-of-plane velocity perturbations. I first fractured the orbits starting
with Dirac's r. The fracturing of the orbit averaged out the binding
relativistic effects and the ending orbit's r was a noticeably little bigger
than Bohr's r. Note that I am saying Dirac's r did not "relax" to Bohr's r.
I next started with Bohr's r itself. As mentioned, the non-perturbed orbit
just sat there at the beginning Bohr's radius. With fracturing the orbit,
the deterministic relativistic effects that kept this orbit stable at Bohr's
r were averaged out, and the final r was noticeably larger than Bohr's r. I
lastly started with the r and v values from eqs. (19) and (20) from my
paper. These are based on a Schwarzschild geometry, but are close to fully
Kerr beginning values. This time the beginning deterministic relativistic
Kerr effects were averaged out, resulting in essentially exactly Bohr's r
value at the end. In simplistic equations, h_bar*dt/dtau ->
(non-relativistic) h_bar with sufficient relativistic-effect-removing
velocity perturbations. Here is a plot of the fractured n = 1 shell:

http://sb635.mystarband.net/fig2.pdf


The n = 1 shell is partially complete (spherical shell looks elliptical
because of perspective and axis scaling). During this entire chaotic motion,
the distance from (0,0,0) that the electron had was insignificantly
different than Bohr's r. I did not force this, the simulation converged on
this value. I had originally thought that it was going to be the frame
dragging effects that would turn things around to Bohr, but in reality it is
more driven by the random perturbations. But fully Kerr is needed, because
only with these initial conditions do the perturbations average out to Bohr
(non-relativistic), in agreement with experimentation. It is the radial
distance that dictates what EM frequency will ionize, and that frequency is
non-relativistic, just as the laboratory data show. The only way to achieve
this prediction was to start with an effectively fully-Kerr state, and
fracture the orbits in a fully electronic Kerr field.

This ties up a lot of loose ends for me. I personally am completely
convinced now, that classic QM/QED/QFT are just too simplistic in their
treatment of electron motion. Is it a particle or is it not? Many say no,
then I ask, what is it at the instant of wave collapse? If a definite
position did not actualize, what does wave collapse mean? The way I've
introduced the stochasticism in this hydrogen monte carlo simulation, could
be interpreted as follows. A definite state is fed into the state propagator
(numerical integrator). A definite state is output. This output state serves
as the mean upon which state perturbations are imposed to get an
actualization of another disturbed, but just as definite final state. This
perturbed state is then fed into the deterministic propagator, etc. Before
the imposition of perturbation, the final state is probabilistically "up in
the air" and when actualized is when "the wave function collapsed." In this
sense, this "stochastic space time" simulation is in somewhat of an
agreement. But QM is not correct, and the world of the atom is much more
chaotically nonlinear (but still quantized) than QM ever dreamed of.

Steve Bell



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