MeV-level-kinetic-energy bound electrons
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A. Meulenberg1, J. L. Paillet2
1
Science for Humanity Trust, Inc., USA
2
Aix-Marseille University, France
Email: Mules333@gmail.com
The relativistic Schrodinger (Klein-Gordon or K-G) and Dirac equations predict deep electron
orbits with radii in the femto-meter range. However, they just predict and use the local total energy,
TE, restmass energy, ME, and binding energy, BE, (TE > 0 and TE - ME = BE < 0) of the
relativistic electrons in these orbits, not their kinetic energies. The kinetic energy (KE > 0) is
determined from the constant sum of all the energies, TE = KE + PE + ME + BE, where ME > 0 is
the rest-mass energy and PE < 0 is the potential energy of the bound electron. At r = , with KE =
PE = BE = 0, then TE = ME. Since 4 of the energies are effected by relativity and only 3 of them
are known, there is ambiguity as to the actual values of both KE and PE. The virial theorem,
relating the KE and PE, provides an extra independent equation for non-relativistic electrons (KE =
|BE| = |PE|/2) and, thus, unique values for all of the energies, as well as that for the emitted photons
of atomic decay (where Eγ = |BE|.). However, for relativistic electrons, the virial theorem also
depends on the velocity so that it can only provide related bounds for the KE and PE values (i.e.,
|PE|/2 < KE < |PE|).
In prior papers, i,ii one of the present authors (AM) assumed the potential energy to be only a
function of distance between the electron and nucleus. He further assumed the values of PE and KE
to be at or near the minimum values that fit the experimental results of cold fusion, which provided
a dominant path for D-D fusion to 4He without the energetic-photon decay (> 20 MeV) of hot
fusion. Such deep orbits and values of KE (in the MeV range) for the bound electron violated the
Heisenberg uncertainty relation (HUR) and was therefore suspect. The second author (JLP) has
subsequently explored the solutionsiii,iv,v with a relativistic potential based on the HUR and found the
femtometer size orbits to have electrons with KE in the 100 MeV range. A companion paper in this
conference continues that development.
This presentation provides further development of the KE = ~ 1 MeV solutions to the relativistic
equations, which must provide non-zero angular momentum values that are far less than the normal
ħ values of atomic orbitals. The basis for the HUR in the quantum mechanics (QM) of orbitals is
explored, along with the relativistic quantum number k that the spin-dependent, relativistic, Dirac
equation introduces. One benefit of this study is that the problem of a singularity at r = 0 is removed
since there is no deep-orbit solution for the zero angular momentum condition (l = 0), which is
required for a wave-function that includes values at r = 0. A second benefit is, hopefully, a better
description and understanding of the nature of orbital resonances that are the basis of relativistic
QM solutions for the near-nuclear-region orbits.
A. Meulenberg and K. P. Sinha, “Tunneling Beneath the 4He∗ Fragmentation Energy,” presented
at 239th ACS National Meeting, San Francisco, CA March 2010, Journal of Condensed Matter
Nucl. Sci., JCMNS, 4 (2011) 241–255
ii
A. Meulenberg and K P Sinha, “Deep-electron orbits in Cold Fusion,” 17th International
Conference on Condensed Matter Nuclear Science, Daejeon, Korea, 12-17 August, 2012, J.
Condensed Matter Nucl. Sci. 13 (2014), 368-377
iii
J. L. Paillet and A. Meulenberg, “Relativity and Electron Deep Orbits of the Hydrogen Atom,”
http://viXra.org/abs/1606.0259
iv
J. L. Paillet and A. Meulenberg, “The Basis for Electron Deep Orbits of the Hydrogen Atom,”
ICCF-19, 19th Int. Conf. on Cond. Matter Nuclear Science, Padua, Italy, 15/05/2015, J. Condensed
Matter Nucl. Sci. 19, pp.230 – 243, (2016)
i
J-L Paillet, A. Meulenberg, “Special relativity: the source of electron deep orbits,” Foundations of
Physics, Vol. 47, Issue 2, pp 256–264, Feb. 2017
v