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Electron-electron bound state
in light of magnetic dipole-dipole interaction
Arayik Emilievich Danghyan
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  .
Keywords: electron-electron pair, magnetic dipole, quantum mechanics, relativistic equation.
Composite electromagnetic potential
The potential energy of magnetic dipole-dipole interaction between two electrons can determined
using formula (1).
( )( )
1 2 1 2
e e e e
r r r
12e e B
an intrinsic magnetic dipole moment, which, as we know, is equal to the Bohr magneton
– the distance between electrons,
– the magnetic constant.
Let us rewrite formula (1), taking into account scalar products and formula (2).
012 1 2
cos 3cos cos
represents the angle between vectors
– the angles between the magnetic moment vectors and the line joining particles.
It is clear that the potential energy (W) depends on the mutual arrangement of dipoles.
The following mutual arrangements of dipoles  are of interest to us (Fig. 1)
a) b) c) d)
Fig. 1 Mutual arrangements of the intrinsic magnetic moments of electrons.
Cases a) and c) represent electron attraction; hence; the energy is negative. Cases b) and d), on the
other hand, represent electron repulsion; therefore, the energy is positive.
It is clear that electron pairing requires attraction. Therefore, we can determine the energy (W) for
cases a) and c) using formula (4), in which case we obtain:
cos cos 0
cos cos 1
Therefore, the attractive potential energy for case c) is twice the value of the energy for case a). Note
that case a) corresponds to oppositely directed electron spins. Therefore, the total spin is equal to 0.
The electron spin direction coincides for case c); therefore, the total spin is equal to 1.
The Coulomb potential energy of interaction between two electrons is represented as follows:
The total potential energy of electromagnetic interaction for case c) can be calculated
c Coulomb e
U W W rr
= + = − +
Let us rewrite formula (5) to make it more convenient. To do this, we will use the well-known relation
, which can be rewritten as follows:
(6). Let us substitute values
(Formula 3) into the potential energy formula (5). We obtain:
2 2 2
2 3 2
() 4 4 4
Ur с r m r
= − +
(7). Let us rewrite formula (7) using Hartree atomic units.
In the Hartree atomic unit system, the following values of physical constants are accepted:
1, 1, 1, 1, 137.03599971,4 1a m e c
= = = = = =
Finally, we obtain the following equation for the
composite potential energy of electromagnetic interaction:
Ur с r r
= − +
(8). Let us graph the
resulting equation (Fig. 2)
Fig. 2 Graph of the composite potential energy of electron-electron interaction.
As we can see, the obtained potential consists of a Coulomb barrier and a deep potential well and has
a characteristic maximum.
Let us determine the coordinates of the maximum potential energy.
in Hartree atomic
units. If we multiply this by the Bohr radius, we will obtain
( ) 74.592950
in Hartree atomic units. If we multiply this value by the Hartree energy, we will
( ) 74.592950*27.2 2028.928244
M2 radial equation for the electron-electron pair
Let us designate the electron-electron pair by the “ee” symbol. The ee pair is a hydrogen-like entity.
And since the masses of these two electrons are equal, it is necessary to replace the mass (m) with the
reduced mass (
) in the equation for the hydrogen atom . Let us write the M2 radial equation
for the ee pair:
2 4 6 22
2 2 2
d R dR c
R c R
dr r dr r E U r
+ − − − =
It is difficult to solve the resulting equation using the analytical methods due to the complex
dependence of the composite potential. Therefore, we will apply a numerical solution method using
FlexPDE software package http://www.pdesolutions.com/.
The numerical solution of the radial equation in the spherical coordinate system
Fig. 3 The radial wave function of the electron-electron pair
Fig. 4 The radial probability density of the electron-electron pair
As a result of the numerical solution of equation (9), the following parameters of the ee pair were
obtained: the orbital radius is equal to 0.134876 pm. Note that this is the distance between the
electrons, between the maxima of the probability density. The binding energy is equal to 265363.7
eV. As we can see, the convergence of the equation is very high, which means that there is a high
probability that this bound state of electrons may actually exist.
The numerical solution of the M2 equation in the cylindrical coordinate system
As we can see, the radial equation (9) includes the angular momentum quantum number
. But, as
we know, it doesn’t include the magnetic quantum number m. Nor does it give us a complete picture
of the spatial configuration of electron shells and the magnitude of the magnetic quantum number m.
This data can be obtained by solving the equation in the cylindrical coordinate system.
The numerical solution of the equation in the cylindrical coordinate system showed us that the
convergence is possible only if the value of the magnetic quantum number is
. We also note that
further research is needed in order to obtain a solution for the orientation of magnetic moments a).
Most importantly, the possibility of existence of a deep bound state of the electron-electron pair has
been theoretically proven.
If this theory is experimentally confirmed, it will be further developed.
Fig. 5 Wave function of the ee pair in the cylindrical coordinate system
Fig. 6 Probability density in the cylindrical coordinate system
Fig. 7 Projection of probability density in the cylindrical coordinate system
Therefore, as a result of the numerical solution of the equation in the cylindrical coordinate system,
the following results have been obtained: the orbital radius is equal to 0.134793 pm. Again, note that
this is the distance between the maxima of the probability density, which is the diameter. The binding
energy is equal to 265363.7 eV. And, as we said, the magnetic quantum number is
. In addition,
we see that the spatial distribution of the electron density is represented by a toroid. However, if we
take a look at the solutions of the radial equation (9) in the spherical coordinate system, we will see
that it resembles a spherical distribution with an empty region inside.
Results and discussions
Simultaneous consideration of the Coulomb electrostatic repulsion and magnetic dipole-dipole
attraction between two electrons gives us reason to believe that a stable bound state may exist when
the electrons approach each other close enough. This is also mentioned by many other authors in their
publications. However, it has not yet been possible to obtain an accurate calculation of all the
parameters for such a state.
In this paper, the possibility of existence of a deep bound state of the electron-electron pair is proved
based on the numerical solution of the M2 relativistic equation. The exact values of the ground state
parameters have been obtained. The orbital radius is equal to 0.1348 picometers. In fact, this is the
diameter, since the equation is set up taking into account the reduced mass. Therefore, the variable
represents the distance between the electrons. The binding energy: 265363.7 electron volts. The total
. The magnetic quantum number:
. The wave functions and probability density
distributions have been obtained in the spherical and cylindrical coordinate systems. It has been
shown that this distribution is represented by a toroid. This characteristic form of the probability
density gives us reason to believe that the ee pair in its ground state is a neutral entity.
In addition to the fact that the above calculations and results are of theoretical interest, they may also
be of great practical value. As shown, the formation of the ee pair leads to the release of energy at
265363.7 eV. This is of great practical value, which means that environmentally friendly power plants
could be built based on this phenomenon.
We think there has been provided enough evidence to conduct an experimental search for an elusive
List of publications:
1. TO THE POSSIBILITY OF BOUND STATES BETWEEN TWO ELECTRONS Alexander
A. Mikhailichenko, Cornell University, LEPP, Ithaca, NY 14853, USA
2. Electromagnetic Theory of the Nuclear Interaction
3. On certain features of electron-proton interaction
Popenko V.I. Scientific and Production Corporation "Kiev Institute of Automation"
4. F. Mayer, J. Reitz, “Electromagnetic Composites at Compton Scale”, arXiv:1110.1134v1
[physics.genph] 10 Sep 2011
5. Solution of the Dirac equation with Coulomb and magnetic moment interactions
Barut, A. O. Kraus, J.
Journal of Mathematical Physics, Volume 17, Issue 4, pp. 506-508 (1976).
6. Two-body problems with magnetic interactions Hesham Mansour, Ahmed Gamal
Journal of Nuclear and Particle Physics 2019; 9(2): 51-58
7. Danghyan A.E. “New equation of relativistic quantum mechanics”
8. Dangyan A.E. “Hydrogen Atom. Exotic states. Part one"
9. Dangyan A.E. “Hydrogen Atom. Exotic states. Part two”