Highly Relativistic Deep Electrons and the Dirac equation

Journal of Condensed Matter Nuclear Science, #29, 2019

We address a number of questions relating to the progress of our study on the relativistic-electron deep orbits (EDOs): (1) How to combine different EM potentials having two possible versions (attractive and repulsive), while rejecting unrealistic energies? (2) What about the angular momentum of the deep electrons? How is the Heisenberg Uncertainty Relation satisfied in these EDOs? (3) From where is extracted the high kinetic energy (of order 100 MeV) of the deep-orbit electrons? (4) What is the behavior of the effective potential V eff as a function of distance to the nucleus? (5) What is the order of magnitude of the radiative corrections for the EDO's? (6) What is the relation between EDO solutions of the Dirac equation and the high energy resonances (with high binding energies) corresponding to a semi-classical local minimum of energy?

This work continues our previous work [1] and in a more developed form [2]), on electron deep orbits of the hydrogen atom. An introduction shows the importance of the deep orbits of hydrogen (H or D) for research in the LENR domain, and gives some general considerations on the EDO (Electron Deep Orbits) and on other works about deep orbits. A first part recalls the known criticism against the EDO and how we face it. At this occasion we highlight the difference of resolution of these problems between the relativistic Schrödinger equation and the Dirac equation, which leads for this latter, to consider a modified Coulomb potential with finite value inside the nucleus. In the second part, we consider the specific work of Maly and Va'vra [3], [4]) on deep orbits as solutions of the Dirac equation, so-called Deep Dirac Levels (DDLs). As a result of some criticism about the matching conditions at the boundary, we verified their computation, but by using a more complete ansatz for the " inside " solution. We can confirm the approximate size of the mean radii <r> of DDL orbits and that <r> decreases when the Dirac angular quantum number k increases. This latter finding is a self-consistent result since (as distinct from the atomic-electron orbitals) the binding energy of the DDL electron increases (in absolute value) with k. We observe that the essential element for obtaining deep orbits solutions is special relativity.

In previous works, we analyzed and countered arguments against the deep orbits, as discussed in published solutions. Moreover, we revealed the essential role of Special Relativity as source of electron deep orbits (EDOs). We also showed, from a well-known analytic method of solution of the Dirac equation, that the obtained EDOs have a positive energy. When including the magnetic interactions near the nucleus, we observed a breakthrough in how to satisfy the Heisenberg Uncertainty Relation (HUR) for electrons confined near the nucleus, in a radial zone of only a few fm. Here we chose a different method, by directly facing the HUR for such confined electrons, from which we deduce the coefficient γ of these highly relativistic electrons. Then we show the effective Coulomb potential due to a relativistic correction, can maintain the electrons in containment. Next we resume and deepen our study of the effects of EM interactions near the nucleus. We first obtain computation results: though approximate, we can effectively expect high-energy resonances near the nucleus. These results should be confirmed by using QFT-based methods.

This work continues our previous works on electron deep orbits of the hydrogen atom. An introduction shows the importance of the deep orbits of hydrogen (H or D) for research in the LENR domain, and gives some general considerations on the Electron Deep Orbits (EDOs). In a first part we quickly recall the known criticism against the EDO and how we face it. In particular, a solution to fix all problems is to consider a modified Coulomb potential with finite value inside the nucleus. For this reason, we deeply analyzed the specific work of Maly and Va'vra on deep orbits as solutions of the Dirac equation, with such a modified Coulomb potential without singular point. Then, by using a more complete ansatz, we made numerous computations on the wavefunctions of these EDOs, allowing to confirm the approximate size of the mean radii ⟨r⟩ of orbits and to find further properties. Moreover, we observed that the essential element for obtaining deep orbits solutions is special relativity. At a first glance, this fact results from an obvious algebraic property of the expression of energy levels obtained by the relativistic equations. Now, a comparative analysis of the relativistic and of the non-relativistic Schrödinger equation allows us to affirm that Special Relativity leads to the existence of EDOs because of the non-linear form of the relativistic expression for the total energy, which implies a relativistic non-linear correction to the Coulomb potential.

In this paper, we look into the difficult question of electron deep levels (EDLs) in the hydrogen atom. Acceptance of these levels and, in particular, experimental evidence of their existence would have major implications for the basis for cold fusion and would open up new fields of femto-physics and-chemistry. An introduction shows some general considerations on these orbits as " anomalous " (and usually rejected) solutions of relativistic quantum equations. The first part of our study is devoted to a discussion of the arguments against the deep orbits and for them, as exemplified in published solutions. We examine each of the principal negative arguments found in the literature and show how it is possible to resolve the questions raised. In fact, most of the problems are related to the singularity of the Coulomb potential when considering the nucleus as a point charge, and so they can be easily resolved when considering a more realistic potential with finite value inside the nucleus. In a second part, we consider specific works on deep orbits, named Dirac Deep Levels (DDLs), as solutions of the relativistic Schrödinger and of the Dirac equations. The latter presents the most complete solution and development for spin ½ particles, and includes an infinite family of DDL solutions. We examine particularities of these DDL solutions and more generally of the anomalous solutions. Next we analyze the methods for, and the properties of, the solutions that include a corrected potential inside the nucleus, and we examine the questions raised by this new element. Finally we indicate, in the conclusion, open questions such as the physical meaning of the relation between quantum numbers determining the deep levels and the fact that the angular momentum seems two orders-of-magnitude lower than the values associated with the Planck constant. Introduction For many decades, the question of the existence of electron deep levels (EDLs) for the hydrogen atom led to numerous works and debates. Why once more a study on this subject? For several reasons: • the arguments in favor of the deep orbits have become progressively more mature by the use of relativistic quantum tools for a full three-dimensional description of the system; • by accepting the reality of a non-singular central potential within a nuclear region, many mathematical arguments against anomalous solutions of the relativistic equations no longer pertain; • numerical evaluation of the relativistic equations are now detailed and available for interpretation of the models, their implications, and their predictions; • hydrogen atoms, including deuterium, with electron deep orbits (femto-atoms) can contribute to the understanding of: processes of cold fusion inside condensed matter, the avoidance of nuclear fragmentation in D-D => 4 He fusion reactions, and a means of increasing the rate of energy transfer between an excited nucleus and the surrounding lattice; • and, above all, recognition of these levels opens up a whole new realm of atomic, nuclear, and subatomic-particle physics as well as nuclear chemistry. There are various theoretical ways to define a state of the hydrogen atom with electron deep level (EDL) or deep Dirac level (DDL) orbits. In the following, we denote H # as any state of hydrogen atom with EDL orbits. Some authors use the term hydrino for denoting the H # states owing to the work of [1] on the hypothetical existence of H atoms with orbit levels under the Bohr ground level and where the values of orbit radii are fractional values of the Bohr radius. Here we do not use this term, a physical concept specifically attached to the cited work, because it is not deduced from (standard) quantum equations, while we essentially consider the states H # obtained by the methods of relativistic quantum physics. With the quantum equations habitually used in the literature for computing the bound states of the H atom, we can note that there is in general a crossroad with a choice of value or a choice of sign for a square root in a parameter. According to which path is chosen, the resolution process leads either to the usual solution or to an unusual one called an "anomalous" solution; one that is rejected in the Quantum Mechanics Textbooks.