These are generous remarks by Dr Gareth Evans and much appreciated by Horst and myself. Gareth is an expert in spectroscopy and optics and has recently done important work with Trevor Morris, scheduled to appear in UFT264. The checking work by Horst Eckardt is indeed of key importance, and on top of that he always produces incisive graphics and new physics in each paper. This has been going on now since about 2007 in about 200 of the 266 UFT papers, and in other articles with Doug Lindstrom and others on www.aias.us.
Sent: 22/07/2014 08:40:10 GMT Daylight Time
Subj: Re: Numerical calculations by Horst Eckardt for UFT266
These are fantastically insightful results demonstrating the power and precision of ECE THEORY yet again.Myron and Horst have developed ways of working and checking results that is unique in the history of science. What a great double act! What a privelege to witness these developments in natural philosophy as they unfold.
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Numerical calculations by Horst Eckardt for UFT266
Many thanks again to co author Horst Eckardt. These are incisive computations carried out in such a way as to show the main conclusions very clearly. The Sommerfeld orbitals are precessing ellipses very close to circles, because the relativistic correction is small. The minimum x was found by computation by Dr. Eckardt to be 0.8. The numerical analysis shows that the Sommerfeld ellipses can be defined only in a very small range around the Bohr radii. This is a new result, which appears to be unknown in the literature. The range of alpha in the Sommerfeld theory is also very restricted, another new result. These results illustrate how x theory is able to deduce new information in quantum mechanics, this time in the context of the old quantum theory. The Bohr and Sommerfeld atoms were chosen to illustrate quantization and relativistic quantization. The Sommerfled theory was the first relativistic quantum theory (1915). The Bohr theory (1913) was essentially heuristic but now x theory shows that it is the limit of an ellipse whose half right latitude gives the Bohr radii for each principal quantum number n, and whose eccentricity gives the energy levels as the former approaches zero (a circle). In the mid eighties at IBM Kingston, New York State the Clementi environment were using LCAP array processors and IBM 3090 and 3084 supercomputers to compute relativistic effects in heavy atoms, using the Dirac equation. Relativistic effects in the hydrogen atom are very small but it is analytical. So x theory based on ECE theory gives a lot of new insight, and I will write up my sections of UFT266 today.
Sent: 21/07/2014 16:18:00 GMT Daylight Time
Subj: Fwd: Numerical calculations for paper 266
I have finished the calculations now according to note 266(10). I think the discussions before were fruitful and clarified a lot of points, at least for me.
The first graph of the attached shows the Bohr circular orbits for n=1 to n=4 in atomic units. Then we simply have r = n^2.
In the subsequent table the relevant values of Sommerfeld theory are shown: x is varied in blocks with n=1 to 4 each. For x=0.8 the square of the ellipticity is negative, giving no meaningful results. According to formula (2) of 266(10) an alpha and epsilon can be attributed to each combination (n, x). We see that these half-right latitudes are very close to the Bohr radii and the ellipticities are very small, in the range of only one thousandth. Therefore the ellipses are practical identical to the Bohr circles in the first figure.
It is interesting to plot the values of x^2 in dependence of varying alpha, see second fig. Here the x^2 values for different n were shifted on the x axis so that they come to lie one upon each other. One sees that there is a pole at alpha = rB or, more precisely, alpha = gamma rB. There is a sharp pole which is even sharper for increasing n. Therefore there is only a very small range around the Bohr radii where the Sommerfeld ellipses are defined, namely in the region with x^2 ~ 1. We see that there the Bohr quantization of orbits is relaxed slightly in Sommerfeld theory but essentially remains valid.
Another plot is alpha(x) in the second Figure. the alpha values have been normalized by r_B so that the y scale is comparable. We see again the stronr restriction of the alpha range which gets even smaller for rising x. As discussed above, there is no ellipse for x<0.8.
The third Figure shows the same behaviour for epsilon(x), This is bounded by an asymptote for each value of n, showing again that Sommerfeld ellipses are extremly close to circles.
I will eventually make a try for varying velocity to see the effect, although it is not physical because other parameters would to be changed too, as we have discussed.