Archive for August, 2015

Plans for UFT326

I intend to continue by using the approach used in previous UFT papers developing the fermion equation, but to explore approximations other than those used by Dirac and contemporaries. In those approximation they used E = gamma mc squared about equal to m c squared in the denominator, UFT247 to UFT253. I like this kind of work because it is so elegant and produces so much information and has so many possible variations on the theme. It is based on the Einstein energy equation, which is a retsatemtn of the realtivistic momentum p = gamma m v (see Marion and Thornton or good websites). It can be looked upon as quantization of the ECE2 Lorentz force equation. In the Einstein energy equation

E squared = c squared p squared + m squared c fourth

where p is the relativistic momentum p = gamma m v. In the Dirac type approximations p is approximated in the development by the non relativistic momentum in the numerator of

E – m c squared = p squared c squared / ( E + m c squared)

After this development I intend to go back to the earlier ECE2 equations and develop them for the spin connection. Dirac used the minimal prescription which is equivalent to adding U as shown in immediately preceding UFT papers. The approximations used by Dirac et al. are crude and rough ones which can only be justified by the agreement with experimental data, the famous half integral spin , ESR, NMR and so on. I suggest strongly that readers follow this discussion with readings of Marion and Thornton, chapter on special relativity. It is especially important to understand the Lorentz transform and the definition of the Lorentz gamma factor from the Minkowski metric. Horst’s demonstration of precession from special relativity in UFT325 is also very important. At first, special relativity can be very confusing but it is not difficult if a few rules are kept clearly in mind.

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Discussion of 326(5), Part Two

The p0 is the Newtonian momentum, so Eqs. (29) and (32) can be solved, p0 does not contain gamma but p contains gamma:

p = gamma p0 = gamma m v = h bar kappa

and
E = gamma m c squared = h bar omega

as in many previous UFT papers.

. The origin of the Lorentz factor gamma is relativistic, i. e.

c squared dtau squared = c squared dt squared – v squared dt squared

Discussion of 326(5)

I agree with these points, for the free particle, E = T, thi sis just a matter of notation. It is clear from page 5 that the velocity appearing in gamma is the Newtonian velocity, see the steps in Eq. (38), page five. I also agree that it does not come from a classical analysis. It comes form the relativistic Minkowski metric as described for example in Marion and Thornton. So the theory is rigorously self consistent and also consistent with the lagrangian theory.

To: EMyrone@aol.com
Sent: 30/08/2015 20:56:45 GMT Daylight Time
Subj: Re: 326(5): Final Version of Note 326(4)

In eq.(15), E is obviously the total energy without rest mass, in contrast to (5).
eq.(24): hbar squared kappa squared
eq.(25): wouldn’t it be better to write mT intead of mE? E is without restmass again here.
eq. (29) allows computing the relativistic kinetic energy if the classical kinetic energy T_0 is known:

T_0 = 1/2 (1+gamma)/gamma^2 T
or
T = 2 gamma^2/(1+gamma) T_0.

However this is not a self-consistent procedure, see below:

Eq.(32) can be written with the classical momentum p_0 but this does not mean that this comes out from a non-relativistic theory. This is rather a “back-transfomation” to a non-relativistic case from a self-consistent relativistic solution of quantum or Lagrange equations. We solved them in paper 325 for example.

Considering the limit gamma –> 1 gives the correct non-relativistic limit, that is o.k.

Horst

Am 30.08.2015 um 15:07 schrieb EMyrone:

I went through my calculations again and found that the correct free particle quantization equation is Eq. (29) with gamma defined by Eq. (32) and the de Broglie wave particle dualism by Eq. (33). So these equations can be solved by computer algebra to give E in terms of p sub 0, the classical momentum, and kappa. The cross check on page (5) confirms that everything is self consistent. Having gone through this baseline calculation the particle on a ring and H atom can be defined in a relativistic context. The answer to the computer algebra must be:

E squared = (h bar kappa c) squared + m squared c fourth

so this gives a check on the results of the computer algebra. The fermion equation for the free particle is therefore Eq. (29) where gamma is given by Eq. (32). and where the de Broglie wave particle dualism is given by Eq. (33). Although these equations look like familiar special relativity they are the quantization of the ECE2 Lorentz force equation.

Daily Report Saturday 29/8/15

There were 1,914 files downloaded or hits from 341 distinct visits or reading sessions, main spiders, baidu, google, MSN and yahoo. Collected Scientometrics 668, F3(Sp) 634, Evans / Morris papers 580 (est), Collected ECE2 papers 488, Autobiography volumes one and two 329, Barddoniaeth / Collected Poetry 310, Eckardt / Lindstrom papers 210. Principles of ECE 187, proofs that no torsion means no gravitation 181, UFT88 136, Evans Equations 123 (numerous Spanish), Llais 98, Engineering Model 96, UFT311 83, CEFE 70, UFT321 61, UFT322 56, UFT320 52, UFT318 52, UFT319 49, UFT324 42, UFT317 42, UFT313 40, UFT316 37, UFT323 37, UFT315 31, UFT314 29, UFT325 21 to date in August 2015. University of the Andes UFT139 (Sp), spidering from a private site in Germany, Istella Media Italy general; Gospel Ministry Alliance general; Izhevsk region extensive. Intense interest all sectors, updated usage file attached for August 2015

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326(5): Final Version of Note 326(4)

I went through my calculations again and found that the correct free particle quantization equation is Eq. (29) with gamma defined by Eq. (32) and the de Broglie wave particle dualism by Eq. (33). So these equations can be solved by computer algebra to give E in terms of p sub 0, the classical momentum, and kappa. The cross check on page (5) confirms that everything is self consistent. Having gone through this baseline calculation the particle on a ring and H atom can be defined in a relativistic context. The answer to the computer algebra must be:

E squared = (h bar kappa c) squared + m squared c fourth

so this gives a check on the results of the computer algebra. The fermion equation for the free particle is therefore Eq. (29) where gamma is given by Eq. (32). and where the de Broglie wave particle dualism is given by Eq. (33). Although these equations look like familiar special relativity they are the quantization of the ECE2 Lorentz force equation.

a326thpapernotes5.pdf

Discussion of Note 326(4): A New Free Particle Relativisic Schroedinger Equation

Many thanks indeed, an exciting discovery! I will proceed immediately to developing these new solutions with Schroedinger quantization in preparation for numerical solution, then add a potential. The quantized versions may be soluble to give completely new relativistic free particle wavefunctions both for translational and rotational free particle motion. The golden age of quantum mechanics always comes up with something new, almost a hundred years after its first development. This is essentially quantization of ECE2 theory. This is of course the method we have used in many previous UFT papers, many variations on a them by Paul Dirac and contemporaries, but always based on geometry so the Dirac equation has become the fermion equation of generally covariant unified field theory (now ECE2 theory).

To: EMyrone@aol.com
Sent: 29/08/2015 11:30:00 GMT Daylight Time
Subj: Exact solution of the relativistic momentum equation

In note 326(4) eq.(33) can be solved with some effort in computeralgebra without approximation v<<c, see attached. There are 4 solutions for p^2, the results differ in signs and a summand m*H1.

Horst

Am 24.08.2015 um 15:57 schrieb EMyrone:

This note uses a Dirac type quantization to produce the equation (21), a relativistic Schroedinger equation which must be solved for the wavefunctions psi. In general this is a highly non trivial procedure which must be carried out numerically in three dimensions. However, it is straightforward to show that this type of quantization produces small shifts in the H atom energy levels given by the expectation value (28). The Thomas factor is given correctly by the Sommerfeld atom, but there is no spin orbit interaction, because the Sommerfeld atom does not contain a spin quantum number, later suggested by the Sommerfeld group itself and developed by Pauli and others. As in previous UFT papers spin orbit coupling and many new effects of development of ECE appears with the use of the SU(2) basis and Pauli matrices. It is known from UFT325 that these orbitals in two dimensions must be the result of a quantization of a two dimensional precessing ellipse, and that will be the subject of the next note. This method is much clearer than that used by Sommerfeld himself in 1913, who did not have the benefit of Schroedinger Debye quantization (circ 1923 / 1924). Sommerfeld produced orbitals in 1913 which he communicated by letter to Einstein.

326(4).pdf

Discussion of 326(4)

Thanks again for going through 326(4). Agreed with the first two points, I thing that the factor 2 is alright because 1 + gamma is approximately 1 + 1 – v squared / (2 c squared).

To: EMyrone@aol.com
Sent: 29/08/2015 11:20:02 GMT Daylight Time
Subj: Re: 326(2): Quantization of the Sommerfeld Hamiltonian

In eq.(39) it should read at the RHS:

hbar^2 / 2 m^2 c^2

(with m squared). In eq.(46) the factor E seems to be missing. Should there be a “3” instead of “2” because of 1+gamma in eq. (30)?

Horst

Am 24.08.2015 um 15:57 schrieb EMyrone:

This note uses a Dirac type quantization to produce the equation (21), a relativistic Schroedinger equation which must be solved for the wavefunctions psi. In general this is a highly non trivial procedure which must be carried out numerically in three dimensions. However, it is straightforward to show that this type of quantization produces small shifts in the H atom energy levels given by the expectation value (28). The Thomas factor is given correctly by the Sommerfeld atom, but there is no spin orbit interaction, because the Sommerfeld atom does not contain a spin quantum number, later suggested by the Sommerfeld group itself and developed by Pauli and others. As in previous UFT papers spin orbit coupling and many new effects of development of ECE appears with the use of the SU(2) basis and Pauli matrices. It is known from UFT325 that these orbitals in two dimensions must be the result of a quantization of a two dimensional precessing ellipse, and that will be the subject of the next note. This method is much clearer than that used by Sommerfeld himself in 1913, who did not have the benefit of Schroedinger Debye quantization (circ 1923 / 1924). Sommerfeld produced orbitals in 1913 which he communicated by letter to Einstein.