Article

# Solution of the Dirac equation with Coulomb and magnetic moment interactions

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## Abstract

The Dirac equation for a charged spin 1/2 particle with an anomalous magnetic moment in the Coulomb field is solved. A new phenomenon of formation of very narrow resonances of very high mass at small distances is demonstrated.

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... Magnetic forces between spin 1/2 particles lead to the effective radial potentials of this type [1][2][3], with one or more deep narrow wells. Magnetic interactions are studied for various problems ranging from macroscopic to microscopic scales [4][5][6][7][8][9][10]. ...
... Let us consider, for example, a relativistic charged spinless particle m in a field of a fixed (quantum ) magnetic moment ⃗ [11], or alterentivly , a charged spin 1/2 particle of mass m and magntic moment ⃗ , in the field of a fixed charge [12,13]. In both cases the radial equation has the form [− 2 2 + − 2 ] ( ) = 0 (1) Where the effective potential is given, apart from the Coulomb potential , by ( ) = 2 + 3 + 4 (2) Clearly, if one solves the same problem with a Dirac equation and give also an anomalous magnetic moment ⃗ to the particle, then additional terms are added to equation (2). Further models may also treat the magnetic moment of both particles. ...
... Let us consider, for example, a relativistic charged spinless particle m in a field of a fixed (quantum ) magnetic moment ⃗ [11], or alterentivly , a charged spin 1/2 particle of mass m and magntic moment ⃗ , in the field of a fixed charge [12,13]. In both cases the radial equation has the form [− 2 2 + − 2 ] ( ) = 0 (1) Where the effective potential is given, apart from the Coulomb potential , by ( ) = 2 + 3 + 4 (2) Clearly, if one solves the same problem with a Dirac equation and give also an anomalous magnetic moment ⃗ to the particle, then additional terms are added to equation (2). Further models may also treat the magnetic moment of both particles. ...
Preprint
In the present work, we present different two body potentials which have oscillatory shapes. The eigenvalues and eigenfunctions are obtained for those problems by solving Schrodinger equation using Nikiforov Uvarov method.
... For critical binding of high-Z nuclei we compare in sect. 5.4 the analytic solutions of the KGP equation to the numerical solutions of DP presented by Thaller [15], and review another analysis done by Barut and Kraus [16,17]. Our findings are summarized in sect. ...
... The DP equation also allows for the so-called superpositronium states as described by Barut and Kraus [16,17]. Such states represent resonances due to the magnetic interaction that reside incredibly close to the center of the atom i.e. r 2 ≈aα /mc, but this feature is absent from the KGP formation of the Coulomb problem as all KGP-Coulomb wave functions which can be normalized can be successfully matched to their Dirac (g = 2) companions. ...
Article
We investigate relativistic quantum mechanics (RQM) for particles with arbitrary magnetic moment. We compare two well known RQM models: a) Dirac equation supplemented with an incremental Pauli term (DP); b) Klein-Gordon equations with full Pauli EM dipole moment term (KGP). We compare exact solutions to the external field cases in the limit of weak and strong (critical) fields for: i) homogeneous magnetic field, and ii) the Coulomb \( 1/r\)-potential. For i) we consider the Landau energies and the Landau states as a function of the gyromagnetic factor (g-factor). For ii) we investigate contribution to the Lamb shift and the fine structure splitting. For both we address the limit of strong binding and show that these two formulations grossly disagree. We discuss possible experiments capable of distinguishing between KGP and DP models in laboratory. We describe the impact of our considerations in the astrophysical context (magnetars). We introduce novel RQM models of magnetic moments which can be further explored.
... Section 5.2 suggests an improved version of the KGP equation, which produces better strong field behavior. For critical binding of high-Z nuclei we compare in section 5.3 the analytic solutions of the KGP equation to the numerical solutions of DP presented by Thaller [13], and review another analysis done by Barut and Kraus [14,15]. Our findings are summarized in section 6 where we also discuss future research directions. ...
... The DP equation also allows for the so-called superpositronium states as described by Barut and Kraus [14,15]. Such states represent resonances due to the magnetic interaction that reside incredibly close to the center of the atom i.e r 2 ≈ aα /mc, but this feature is absent from the KGP formation of the Coulomb problem as all KGP-Coulomb wave functions which can be normalized can be successfully matched to their Dirac (g = 2) companions. ...
Preprint
Full-text available
We investigate relativistic quantum mechanics (RQM) for particles with arbitrary magnetic moment. We compare two well known RQM models: a) Dirac equation supplemented with an incremental Pauli term (DP); b) Klein-Gordon equations with full Pauli EM dipole moment term (KGP). We compare exact solutions to the external field cases in the limit of weak and strong (critical) fields for: i) homogeneous magnetic field, and ii) the Coulomb 1=r-potential. For i) we consider the Landau energies and the Landau states as a function of the gyromagnetic factor (g-factor). For ii) we investigate contribution to the Lamb shift and the fine structure splitting. For both we address the limit of strong binding and show that these two formulations grossly disagree. We discuss possible experiments capable of distinguishing between KGP and DP models in laboratory and note that novel RQM models of magnetic moments can be explored.
... We assume that the a k can be so defined. This system is a generalization of one used in [1] to model a relativistic electron in a Coulomb field with anomalous magnetic moment; in [1] the weights were set to unity. For (3.8) we see that ...
... We assume that the a k can be so defined. This system is a generalization of one used in [1] to model a relativistic electron in a Coulomb field with anomalous magnetic moment; in [1] the weights were set to unity. For (3.8) we see that ...
Article
In this paper we consider the one dimensional Dirac system 1.1 where α k (x) < 0, λ is a complex spectral parameter, and the remaining coefficients are suitably smooth and real valued. We regard (1.1) as regular at x = a but singular at x = b ; in Section 4 we extend our result to problems having two singular endpoints. Equation (1.1) arises from the three dimensional Dirac equation with spherically symmetric potential, following a separation of variables. For the choices p(x) = k/x , α k (x) = 1, p 2 ( x ) = (z/x) + c , p 1 ( x ) = (z/x) – c , and appropriate values of the constants, (1.1) is the radial wave equation in relativistic quantum mechanics for a particle in a field of potential V = z/x [ 17 ]. Such an equation was studied by Kalf [ 11 ] in the context of limit point-limit circle criteria, which is one of the matters we consider here.
... The existence of bound states of neutrino (latent) with protons, deuterons, and other nuclei follows from the well-known estimations of anomalous neutrino magnetic moment 1 and the Dirac's equation. 2,3 The concept of relic neutrino leads to the possibility of the neutrino component of the matter. If we assume the existence of the neutrino component of the matter, the question arises as to whether these neutrinos are capable of initiating nuclear transmutations. ...
... 6 The assumption on the occurrence of reactions (1)-(3) appeared as a result of the study of the background gamma spectra formation in germanium gamma spectrometers, used in astrophysical research. 4 Given below is experimental evidence on the existence of reactions (1)- (3). The data proving the existence of the above reactions are also provided ...
Article
Anomalous elemental changes have been observed on the Pd complexes after D2 gas permeation. This effect--effect Y. Iwamura--belongs to a new category of nuclear reactions. The effect of Y. Iwamura can stimulate development of physics of electromagnetic interaction neutrino including physics of relic neutrino and physics of the dineutrons. It is possible to suggest that low-energy neutrino and even relic neutrino can initiate effect of transmutation in special cases. The suggested hypothesis application about new class nu- nuclear reaction existence can be useful for the problems: alternative energetic, radioactive isotopes reducing and rare isotopes production.
... where q, p i , r ij ∈ C ([a, b]) (i, j = 1, 2) are complex-valued functions, and E ∈ C is in general a complex parameter. Apart from few cases in which the Dirac equation possesses exact solutions given in a closed-form (see, e.g., [15][16][17][18]), numerical methods are available for calculating approximate solutions and estimating parts of the spectra [19][20][21][22][23][24]. Nonetheless, the recent paper [25] provides exact solutions to system (14a)-(14b) in the form spectral parameter power series (SPPS). ...
Article
Full-text available
In this work we consider the one-dimensional Dirac equation including an electrostatic potential with compact support, and focus on the regime of bound states. We obtain exact expressions for both the characteristic function and the eigenfunctions in L ² (ℝ, ℂ ² ), given in the form of power series of the energy parameter. This approach is meant for arbitrary bounded potentials, so that a square potential is a special case of the theory here presented. We derive an efficient numerical method for the calculation of approximate eigen-energies of the bound states. Finally, we investigate the physical sense of the eigen-energies that are forbidden in the non-relativistic regime in terms of the Klein tunneling.
... The Vigier-Barut (V-B) model and the works related to this model derived from works of Barut, e.g., Refs. [35,36], which were carried in a relativistic context, with a more complete Dirac equation. However, the above-named V-B model and the related works were made in a nonrelativistic framework. ...
Chapter
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We look into the difficult question of electron deep orbits (EDOs) in the hydrogen atom. Introductory chapters show these orbits as “anomalous” (usually rejected) solutions of relativistic quantum equations; their interest for LENR; the principal negative arguments found in the literature; how it is possible to resolve the questions raised; and some specific works on EDOs. We analyze the EDO works of Maly and Va’vra and define a more complete ansatz for the “inside” solutions. We demonstrate that the essential element for obtaining EDOs is Special Relativity. To extend the model, we analyze the magnetic interactions near the nucleus, with the aim of solving important physical questions: do the EDOs satisfy the Heisenberg Uncertainty relation (HUR)? Are the orbits stable? Finally, we demonstrate how to satisfy the HUR for electrons confined near the nucleus (relativistic effects on the nuclear potentials), and why we expect high-energy resonances near the nucleus.
... (c) E-mail: pinaki.roy@tdtu.edu.vn (corresponding author) the relativistic Coulomb problem in a higher-dimensional curved space of constant positive [12,13] as well as negative curvature [14] have been found. In view of the observations made earlier, we feel it is important to study the Dirac Coulomb problem in (2 + 1)-dimensional curved spaces of constant positive and negative curvatures. ...
Article
In this paper, we study generalizations of the two-dimensional relativistic Coulomb problem in curved geometries with constant positive and negative curvature. It is shown that in both cases the effective Schrödinger-like equations exhibit features of broken supersymmetry and the spectrum is obtained using the SWKB method. Some features of the spectra and restoration of supersymmetry in the zero curvature limit have also been analyzed.
... First, we analyzed several works on this subject from Barut [10][11][12][13] as well as the subsequent works on the socalled "Barut-Vigier model" [14][15][16][17][18] on the hydrogen atom. These latter papers were developed in a non-relativistic context, unlike those of Barut himself. ...
Preprint
Full-text available
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.
... The possible structure of the nuclear molecules that was observed in this experiment can be explained with the magnetic beads model. This model is in agreement with model where nuclear molecules are formed fro m Vigier ato ms [5][6] coupled by magnetic mo ments in a linear thread-like structures. It is assumed that in nuclear molecu le some nucleon transfers are possible. ...
Conference Paper
Full-text available
Several self-consistent observations have led to the conclusion that long-lived multi-core systems (nuclear molecules) exist. This conclusion is based on next results: 1. Some even and no add Bi isotopes were observed in the experiment; 2. The large-scale generation of elements in the clusters spontaneously emitted from the bismuth salt samples was found. There was no bismuth in these clusters, but only possible bismuth decay products (carbon, potassium) 3. Bismuth isotopes generation and clusters emission appeared with more that one year delay; 4 Bismuth isotopes generation and clusters emission might be initiated by flash light or alpha irradiation; 5. Emitted clusters had magnet properties; 6. The traces of the long linear structures (nuclear molecules) movement on the Si detector surface were found. The phenomenological model of the nuclear molecules was suggested.
... The Vigier-Barut model and the works related to this model, derive from works of Barut, e.g. [23,24]. In these articles, the author looks for an analytic solution of the Dirac equation for a charged lepton with anomalous magnetic momentum (AMM) in Coulomb potential. ...
Conference Paper
Full-text available
In the previous works, we discussed arguments for and against the deep orbits, as exemplified in published solutions. So we considered the works of Maly and Va'vra on the topic, the most complete solution available and one showing an infinite family of EDO solutions. In particular, we deeply analyzed their second of these papers, where they consider a finite nucleus and look for solutions with a Coulomb potential modified inside the nucleus. In the present paper, we quickly recall our analysis, verification, and extension of their results. Moreover, we answer to a recent criticism that the EDOs would represent negative energy states and therefore would not qualify as an answer to the questions posed by Cold Fusion results. We can prove, by means of a simple algebraic argument based on the solution process, that, while at the transition region, the energy of the EDOs are positive. Next, we deepen the essential role of Special Relativity as source of the EDOs, which we discussed in previous papers. But the central topic of our present study is an initial analysis of the magnetic interactions near the nucleus, with the aim of solving important physical questions: do the EDOs satisfy the Heisenberg Uncertainty relation (HUR)? Are the orbits stable? So, we examine some works related to the Vigier–Barut Model, with potentials including magnetic coupling. We also carried out approximate computations to evaluate the strength of these interactions and the possibilities of their answering some of our questions. As a first result, we can expect the HUR to be respected by EDOs, due to the high energies of the magnetic interactions near the nucleus. Present computations for stability do not yet give a plain result; we need further studies and tools based on QED to face the complexity of the near-nuclear region. For the creation of EDOs, we outline a possibility based on magnetic coupling.
... • To study the stability of EDOs, we still have to work more deeply on the properties of magnetic interactions and other possible effects near the nucleus, in order to evaluate the possible combinations of potential energies. In particular, the ones involved in the works of Vigier [32], Barut et al. [33] and Samsonenko et al. [34], and the correction to the Dirac operator due to the anomalous magnetic moment of the electron [35] might pertain. ...
Conference Paper
Full-text available
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.
... Both electron and positron have large magnetic moments which contribute to the second potential well in effective potential, at distances much smaller then Bohr radius. Barut and his coworkers predicted that this second potential well can support resonances [3,4]. ...
Article
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... Critchtield (1976) has generalised the Dirac equation in a central field to include the scalar potential proportional to r and r -1. Barut and Kraus (1976) have solved the Dirac equation with the Coulomb potential plus the additional interaction due to the anomalous magnetic moment of the electron in the Coulomb field. The aim of this paper is to present the exact solutions for the electromagnetic potentials which assume particular functional dependance on the space co-ordinate. ...
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In this paper Dirac equation for two electromagnetic potentials viz vector potential and scalar potential have been solved. These solutions of the Dirac equation are written in terms of known solutions of the Schrödinger equation. The presentation is within the two-component relativistic description. Mainly the bound state solutions have been obtained.
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There are many publications out there dealing with the problem of magnetic interaction between elementary particles with intrinsic dipole moments. Basically, the magnetic interaction becomes significant at sufficiently small distances; therefore, the problem is complicated by the need to take into account relativistic effects. The derived equations with the composite potential of the Coulomb and magnetic dipole-dipole interactions generally do not have a clear and simple analytical solution. In this paper, we propose an approach to study a particular case of electron-electron interaction by numerically solving the M2 equation [7] [9].
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Chapter
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Chapter
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Chapter
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Chapter
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We present exact solutions of the Dirac equation for two electromagnetic potentials, i.e. the vector and the scalar potentials. These solutions are written in terms of the known solutions of the Schrödinger equation. The presentation is within the two-component relativistic description. Mainly the bound state solutions have been obtained.
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The 16 × 16 spinor equations of two fermions interacting with their charges and anomalous magnetic moments have been separated first covariantly into center of mass and relative coordinates, then completely into angular and radial parts. The 16 radial equations reduce by a symmetry into two sets of 8 equations, four of which are algebraic. The final result is a set of four first order equations or two coupled second order equations.
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We derive relativistic equations for two-fermion systems from quantum field theory, taking into account the form factors of the particles. When theq 2-dependence of the form factors is disregarded, we obtain, in the static approximation, the two-fermion equations with Coulomb and anomalous-magnetic-moment interactions. Separating the angular variables, we finally obtain a sixteen-component relativistic radial equation.
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Upper and lower limits for the number of bound states in a given central potential are obtained. They imply that for strongly attractive potentials the number of bound states of given angular momentum increases as the square root of the strength of the potential.
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We have solved the Dirac equation for a lepton with an anomalous magnetic moment in the Coulomb field of the antilepton. We demonstrate the existence of resonance states of hadronic size and of energies of a few GeV due to the interplay of effective attractive and repulsive dynamical potentials of the r-2, r-3 and r-4-type. No new parameters are involved. We predict almost similar resonances in the lepton-lepton systems for opposite values of the quantum number kappa.