407(4): Development of the New Theory of Atoms

407(4): Development of the New Theory of Atoms

All is OK here I think because only the expectation value < v squared> is actually used. This is worked out from < p squared / (2m)> . after working it out it is used inside the gamma factor. So the use of the wording "expectation value of gamma" should be avoided. agreed with the second point. The calculation needs the expectation value <1 / r > which you have worked out in previous papers. For n = 1 it is the Bohr radius. Agreed about the virial theorem. This is the first inroad to the relativistic theory. As usual some checking and development work will follow.
407(4): Development of the New Theory of Atoms
To: Myron Evans <myronevans123>

Can the expectation value (statistical average value) of gamma in eq.(16) be transferred to <v^2> ? This is not a linear transformation, and for any function f(v) we have

< f(v) > not equal f( <v> )

because f weights the values of v differently.
In (28) the factor <1/r^2> seems to be missing. Isn’t this the Bohr radius for the H atom, at least for n=1?

Perhaps another point to consider: The so-called Virial theorem:

<T> = – 1/2 <U>.

Normally this only holds for certain forms of the potential energy (type 1/r^n) but only in the non-relativistic case. This theorem is fulfilled here.

Horst

Am 13.05.2018 um 13:49 schrieb Myron Evans:

407(4): Development of the New Theory of Atoms

The major new insight is given by Eq. (1), which is developed into a relativistic theory. This gives several new insights on the non relativistic and relativistic levels. For example the energy levels of the Schroedinger H atom are the negative of the expectation values of the electronic kinetic energy in each orbital. The energy levels of the Sommerfeld atom are the Thomas precession per radian multiplied by the rest energy, m c squared. The force equation and spin connection are derived, and the expectation value of the Lorentz factor defined. The ordinary non relativistic kinetic energy is the Thomas half multiplied by the rest energy. So elements of relativistic theory combine to give a familiar non relativistic result. The Thomas half is obtained by spinning the ECE2 line element.

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