Yes, just to be totally confusing the two books use different notation and in the final UFT267 the notation used in previous papers can be used to avoid this confusion. These results are full of interest as usual. The most fascinating result to me is that the expectation values for the phi plane are all the same, for every orbital, and are each corrected for ellipticity by the same factor. For circular Bohr orbitals this factor is unity. However the classical hamiltonian gives a two dimensional ellipse, not a circle, and the quantum results must also be based on the ellipse. Even more accurately it must be a precessing ellipse of x theory and going in to that will be my next task. In the theta plane the results are completely different as you show. Nothing like this appears in any of the textbook literature on the Schroedinger equation. So by a scholarly inspection of the fundamentals a very large amount of completely new results comes out very often in many UFT papers. By now Horst has the code set up in entirety for the H atom and for each calculation the Born normalization is checked and found to be correct. An orbital of H is three dimensional, but the hamiltonian gives a two dimensional ellipse which must be embedded in three dimensions as discussed here. There are an infinite number of ways of doing this, and the computation gives results from the theta plane and phi plane. These are completely different sets of results, unloading a huge amount of new results on an unsuspecting world. So there is a lot more to the hydrogen atom than ever dreamed of. A new subject area of computational quantum chemistry is opened up for exploration by these results. Although H looks complicated it is in fact peanuts for contemporary computational quantum chemistry, which can handle DNA and superstructures of molecules and so on. This type of result will appear for every atom and molecule and in general across all of computational quantum chemistry. This was the Clementi group speciality at IBM Kingston in the mid eighties, using experimental array processing and supercomputers. Today’s desktops can crunch out a tremendous amount of new data very quickly, as Horst shows here So supercomputers and code packages can be used now for this type of information that will characterize all atoms and molecules. So it looks as if x theory is a goldmine of new information in all directions, as of course is ECE theory.
Sent: 28/07/2014 21:36:24 GMT Daylight Time
Subj: Re: 267(4): Expectation Values of rI have to remark first that the Vector Analysis Problem Solver denotes the angles just inversely compared to the standard denomination in quantum chemistry which is used by Atkins. In the latter, phi is the azimuth angle and appears in form of terms exp(i m phi). I highly recommed to keep this definition because we used it in earlier papers and my code is based on this.
We have two possibilities of symmetrically placing the elliptic orbits in a 3D coordinate system: in the phi plane and in the theta plane. The results of both are given in the attached program output. Placing the ellipse in the phi plane gives the result you obtained. Since the phi factor in the spherical harmonics is always the same, the result is independent of the (l,m) quantum numbers, as comes out from the calculation.
Putting the ellipse into the theta plane gives quite different results. They now depend on the quantum numbers and give logarithmic expressions, indenpendent of principal quantum number n, because the spherical harmonics do not depend on n.Horst
Am 28.07.2014 13:29, schrieb EMyrone
This note gives the expectation values of r from x and Schroedinger quantization. The results are different from Bohr theory in general and it will be very interesting to calculate them for higher order orbitals by computer algebra, using code adapted from pervious UFT papers.