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Highly relativistic deep electrons and the Dirac equation (Note: to be published in JCMNS)

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After analyzing, in the literature, deep orbit results of relativistic quantum equations, we studied them in a semi-classical way, by looking for a local minimum of total energy of an electron near the nucleus, while respecting the Heisenberg Uncertainty Relation (HUR). Now, while using new information thanks to semi-classical computations, we come back to deep electrons as solutions of the Dirac equation, to solve several important and subtle outstanding issues, such as the continuity of derivatives of wavefunctions, a spectral problem about the energy levels associated with the wavefunctions to compute, as well as essential relativistic and energy parameters of the solutions. We thus obtain a better completeness of the solutions. Finally, we give some approaches on the probability of the presence of Electron Deep Orbit (EDO) states in H atom. Introduction Our works on the Electron Deep Orbits (EDOs) are motivated by the need to develop and complete a theoretical model to explain some of the outstanding questions about low-energy nuclear reaction (LENR) results. These results, such as the quasi-absence of high-energy radiation and ejection of particles, require an understanding of the nuclear processes involved [1][2] as well as the means of influencing them from a lattice. Moreover, a better understanding of EDOs and their interaction with nuclear fields will hopefully lead to a practical means of populating these deep levels in a nuclear region from which they can alter nuclear properties (e.g., transmutation and nuclear-decay processes [3]) and facilitate electron capture into the nucleus.
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1
Highly relativistic deep electrons and the Dirac equation
!
#Jean-Luc Paillet1, Andrew Meulenberg2
1Univ. Aix-Marseille, France, jean-luc.paillet@club-internet.fr
2Science for Humanity Trust, Inc., USA!
!
Abstract. After analyzing, in the literature, deep orbit results of relativistic quantum equations, we studied them
in a semi-classical way, by looking for a local minimum of total energy of an electron near the nucleus, while
respecting the Heisenberg Uncertainty Relation (HUR). Now, while using new information thanks to semi-
classical computations, we come back to deep electrons as solutions of the Dirac equation, to solve several
important and subtle outstanding issues, such as the continuity of derivatives of wavefunctions, a spectral
problem about the energy levels associated with the wavefunctions to compute, as well as essential relativistic
and energy parameters of the solutions. We thus obtain a better completeness of the solutions. Finally, we give
some approaches on the probability of the presence of Electron Deep Orbit (EDO) states in H atom.
Keywords:**Deep$electron$levels,$LENR,$Dirac$equation,$Heisenberg Uncertainty Relation, Relativistic effects
Introduction
Our works on the Electron Deep Orbits (EDOs) are motivated by the need to develop and complete a
theoretical model to explain some of the outstanding questions about low-energy nuclear reaction
(LENR) results. These results, such as the quasi-absence of high-energy radiation and ejection of
particles, require an understanding of the nuclear processes involved [1][2] as well as the means of
influencing them from a lattice. Moreover, a better understanding of EDOs and their interaction with
nuclear fields will hopefully lead to a practical means of populating these deep levels in a nuclear
region from which they can alter nuclear properties (e.g., transmutation and nuclear-decay processes
[3]) and facilitate electron capture into the nucleus.
Over the last 3 years, we have analyzed results based on the use of relativistic quantum equations,
because it was to be expected that an electron required to mediate fusion of two nuclei must maintain its
high-probability of being between them and therefore must be relativistically or otherwise confined.
That is why we particularly analyzed and extended [4] the results of [5] and validated [2] the need for
Relativity.
In our more recent works, we took the question of the EDO from a different angle, by studying in a
semi-classical way, the possibility of a local minimum of total energy for an electron in the vicinity of
the nucleus. For this, we consider combinations of attractive and repulsive potentials [6], as well as the
action of radiative corrections, such as the Lamb shift, while satisfying both the Heisenberg Uncertainty
relation (HUR) [7] [8] and the virial theorem. Facing for the first time the thorny question of the HUR
for electrons confined in deep orbits, we were able, not only to evaluate the coefficient
γ
of these highly
relativistic electrons, but also to show that a strong relativistic correction to the Coulomb potential leads
to an effective potential capable of confining such energetic electrons.
In the present work, being equipped with these new insights and methods, we come back to a important
and subtle theoretical question encountered during initial EDO calculations with the Dirac equation,
which showed a significant overlap of the electron wavefunction with the physical nucleus: Should the
energy levels, usually obtained with the Dirac equation solved while considering a point-like nucleus,
be modified and how? In fact, computation of the energy of a deep-orbit electron from its probability-
density distribution, allows us to adjust its initial energy level by applying a fixed-point method.
Moreover we improve the semi-analytical solutions of the radial equations, to obtain wavefunctions
having continuous derivatives on the femto-meter scale, including the surface delimiting the inside and
outside of the nucleus. Doing this, we study how to preserve the initial coupling between the two
components of Dirac solutions for EDOs. Finally, we give some approaches on the question of
populating EDO states.!
2
I. Initial EDO results, as solutions of Dirac equation for atom H.
$
1.1. The anomalous solutions of the Dirac equation.
We had analyzed specific works of Maly and Va’vra on deep orbits as solutions of the Dirac equation.
These orbits were named by those authors "Deep Dirac Levels" (DDLs). They present the most
complete solution and development available, including an infinite family of EDO solutions [9] for
hydrogen-like atoms.
The Dirac equation for an electron in the central “external” Coulomb field of a nucleus, can take the
following form:
𝑖𝜕!+𝑖𝑐 𝛂 .𝛁β 𝑚𝑐!𝑉Ψ𝑡,𝐱=𝟎$$$$$$$$$$$$$$$$$$$$$$$$(1)!
where 𝛂 and β represent the Dirac matrices, 𝛂 is a 3-vector of 4X4 matrices built from the well-known
Pauli matrices, and V is the Coulomb potential, defined by e2/r.
During the solution process with an ansatz, the following condition must be satisfied by a parameter
occurring in the ansatz: s = ± (k2 -
α
2)1/2. The scalar
α
represents the coupling constant (not to be
confused with the vector of Dirac matrices α occurring in the Dirac equation above). If taking the!
positive sign for s, one has the usual “regular” solutions for energy levels, whereas with the negative
sign, one has the so-called “anomalous” solutions. The general expression obtained for the energy levels
of atom H is the following:!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!𝐸=𝑚𝑐!1+!!
!!!!!
!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(2)!
So, while considering the “anomalous” solution, with negative s, the expression of E reads
𝐸=𝑚𝑐!1+!!
!!!!!!!!
!
!!
!
(3)
where n’ is the radial quantum number while k is the specific Dirac angular quantum number, which can
take any integer value 0. In fact, all solutions expressed by E do not correspond to deep orbits, but
only those satisfying the relation n’ = |k| (note that k can be <0). Indeed, we can see that if |k|= n, the
sub-expression D occurring at the denominator of the expression E, D = n’- (k2 -
α
2)1/2, becomes D = n
- (n2 -
α
2)1/2, which is very small since D ~
α
2/2n’, and so E ~ mc2
α
/2n. Then |BE| ~ mc2 (1
α
/2n’)
and |BE| is close to the rest mass energy of the electron, 511 keV. Note, that since k cannot equal 0, then
neither can n.
$
Here, we summarize some features of the “anomalous” solutions of the Dirac equation [6]
• If |k|> n’, the solutions correspond to negative-energy states
If |k| = n’, these special solutions correspond to positive-energy states, and they are the only
ones providing EDOs. Moreover, we can observe, in the energy tables in [9], the following property:
the binding energy in absolute value, |BE|, increases when n’ increases. This is a behavior opposite to
that of the “regular” states. Note that one can also directly deduce this property from the algebraic
expression of |BE| ~ mc2 (1
α
/2n’)
!!If |k| < n’, each solution corresponds to a positive energy state, but E is very close to the energy
of a regular level corresponding to a value of the principal quantum number N taken equal to n’- |k|.
1.2. The deep orbits, as solutions of the Dirac equation with a corrected potential for a nucleus of
finite size.
In a second work [5], the authors determine the wavefunctions of EDOs for hydrogen-like atom
solutions of the Dirac equation. But this time, they consider the nucleus not to be point-like, and thus
the potential inside the nucleus is finite at the origin r = 0. We have seen, in the previous works on
criticisms [10] [11], that this allows eliminating the problems related to the singularity of the classical
Coulomb potential in 1/r.
3
To solve the Dirac equation, the authors use results from Fluegge [12]. As usual, the process includes
separate angular and radial variables and leads to a system of coupled first order differential equations
on both radial functions f(r) and g(r). In the method used by Fluegge, the equation system is
transformed into a 2d order differential equation, a Kummer's equation, and the general solutions of this
equation take the following form, with confluent hyper-geometrical series requiring suitable
convergence conditions:
(4)
!
Note: The parameters a and
µ
include the energy E defined by the expression (3). For example, we have
µ
= [(mc2- E)/(mc2+ E)]1/2 ; likewise, p and q are defined by means of
µ
, so they depend on E too.
!
To solve the equation with a nucleus of finite size 0, the authors carry out the following steps:
- To choose a radius R, the so-called “matching radius”, delimiting two spatial domains: an!
“outside” one, where the potential is correctly expressed by the usual 1/r Coulomb potential, and an
“inside” one, where the potential cannot be expressed by the Coulomb potential. Of course, this choice
may seem arbitrary, but it takes physical meaning if one chooses a value R close to the “charge radius”
Rc of the nucleus.
- To choose a “suitable” expression for the “inside potential”. It is again an arbitrary point, but
we observed [4] this choice has weak influence on the numerical results that interested us, especially the
value of mean radius as function of the radial number n’.
- To satisfy continuity conditions at the "matching radius" R for connecting the inside and
outside potentials. The potential chosen by the authors is derived from the Smith-Johnson potential,
corresponding to a uniformly distributed spherical charge, whose expression is the following:
𝑉(𝑟)=!
!
!
!
!!
!! ! !!
! (5)
where Z = 1 for atom H, and e is the electron charge.
- To solve the system of radial equations for the “outside potential”, i.e. Coulomb potential, that
gives the outside solution composed of two components: functions fo and go. Here, the “outside”
functions fo and go are respectively the functions f and g expressed above while choosing s < 0, i.e.
s= (k2 -
α
2)1/2, to have “anomalous” solutions and by putting |k| = n to discriminate the special
solutions corresponding to EDOs.
1.2.1. Ansatz used for finding the “inside” solutions and continuity conditions.
The choice of ansatz is a very important element for finding fi and gi, solutions inside the nucleus, of the
system of radial equations. Moreover its expression is determinant to satisfy the continuity condition. A
complete analysis of this question is given in [4].!
In their paper [5], the authors put the ansatz in the following form:
𝑔!=𝐴𝑟!!!! 𝐺!𝑟
𝑓
!=i𝐵𝑟!!!! 𝐹
!(𝑟)
where 𝐺!𝑟 and 𝐹
!(𝑟) are in principle power series, i.e.
𝐺!𝑟 = a1 r + a2 r2+ a3 r 3 + … and Fi(r) = b1 r + b2 r2 + b3 r3 + …,
But one may consider approximations of these series by polynomials, by taking into account the
following facts:
- fi and gi must be defined for r < R
- For r < R, very small, the higher-power terms vanish as the degree increases.
!
g=1
2Cr s1er/a
1F
1s+p, 2s+1;2 r
a
"
#
$%
&
's+p
k+q1F
1s+p+1, 2s+1; 2 r
a
"
#
$%
&
'
(
)
*
+
,
-
f=i
2
µ
Cr s1er/a
1F
1s+p, 2s+1;2 r
a
"
#
$%
&
'+s+p
k+q1F
1s+p+1, 2s+1; 2 r
a
"
#
$%
&
'
(
)
*
+
,
-
4
The classical method used, after inserting the ansatz into the equations, allows one to determine the
exponent si and the polynomial coefficients, in order to obtain the solutions. This requires solving a
couple of interdependent recurrent formulas for computing the coefficients of both power series 𝐺!𝑟
and 𝐹
!(𝑟). Nevertheless, it seems the information given in the paper is incomplete, or more precisely,
the chosen ansatz is not complete and it does not contain enough free parameters to satisfy the
continuity condition for both couples of functions (fi, gi) and (fo, go) in R. In fact, useful information was
included in another paper by the same authors, referenced as “to be published,” but never published.
To resolve this problem, we looked for a more complex ansatz including an additional free real
parameter λ, necessary to connect in a suitable manner the inside and outside functions, where the
series/polynomials have the following form:
𝐺!𝑟 = a1 (λ r) + a2 (λ r)2+ a3 (λ r) 3 + … and F
i
(r) = b1 (λ r) + b2 (r)2 + b3 (λ r) 3 +
The continuity conditions {gi(R) = go(R), fi(R) = fo(R)} lead to a system of two algebraic equations. We
showed in [$ P&M, Toulouse] that, for any degree n of the polynomials, the maximum power of λ in
this system of equation remains constant and equals 2, and so this system provides suitable solutions.
1.2.2. Computing the orbital mean radii
Summarily, the computation process for mean orbit radius for a given value of n’ includes the following
steps:
- To determine both couples (fo, go) and (fi, gi) of respective outside and inside solutions. At this
step, the four functions fo, go, fi, and gi include parameters still to be determined
- To connect them in a suitable manner, e.g., by satisfying the continuity conditions, in order to
obtain a couple of “global” wavefunction solutions (F,G). During this step, the unknown parameters
included in the initial functions fo, go, fi, and gi are fixed. The functions, thereby completely defined, can
be denoted by Fo, Go, Fi, and Gi
$- To compute the normalization constant N by using the following formula:
1/N!=! 𝐸𝑙𝐷𝑖 𝑑𝑟 +𝐸𝑙𝐷𝑜 𝑑𝑟
!!
!!
!!
!!
where ElDi represents the electron probability density corresponding to the couple of inside functions
(Fi, Gi):
ElDi = 4π r2(|Fi|2+|Gi|2)
!- Finally, to compute the mean radius <r> by using the following formula:
<r>!=!𝑁 [𝑟 𝐸𝑙𝐷𝑖 𝑑𝑟 +𝑟 𝐸𝑙𝐷𝑜
!!
! 𝑑𝑟
!
!]!
1.2.3. Obtained results and discussion on imperfections.
We give some examples of wavefunction solutions computed for the hydrogen atom H, while using the
following choices
- R = 1.2 F,
!- A nuclear potential defined by the expression (5) given previously, where the proton is
approximated by a uniformly charged solid sphere,
- The polynomials of our ansatz have degree 6.
!
In Fig. 1, we plot the normalized electron probability density functions (NEPD) for n’=1, 2, and 3. The
peak values for NEPD correspond to r ~ R.
5
Fig.1. NEPD, for n’=1 (blue), n’=2 (red), n’=3 (green), with n'=|k|. The radius ρ is in F
Values of mean radius <r> and total energy E for n’=1, 2, and 3.
Note that in these computations, E is deduced from eq. 3, which gives E ~ mc2
α
/2n’.
n'=1, <r>~ 6.6 F, E ~ 1.86 keV
n'=2, <r>~ 1.7 F, E ~ 0.93 keV
n'=3, <r>~ 1.4 F, E ~ 0.62 keV
Now, we can make the following remark: to find out how to populate deep levels, an essential and
concrete question about utility of EDOs for LENR, we need to know more information and to correct
some imperfections. These are listed here in the form of three problems, in an order that has no
significance of importance.
Problem #1. The ansatz we had used does not allow us to have continuous derivatives at the
connection radius R. This problem is more serious than it appears at first sight. The initial Dirac
equation and the resulting system of two radial equations, after separating the variables, are 1st order
equations. Nevertheless, the radial equations are not independent (see e.g. [13]) but interdependent, as
both components f and g occur in each equation. In fact the system is equivalent to a differential
equation of 2d order, and it is completely solved by using effectively a 2d order equation, e.g. a
Kummer's equation [14] or a Whittaker's [15] equation. Under these conditions, it is mathematically
necessary for the global solution function to be not only continuous, but also to have a continuous
derivative everywhere in the domain of real numbers. In [16], we have obtained this result for the "large
component", but it is an approximation, and it should be extended to both components to have better
information on the wavefunctions.
!• Problem #2. The expression (4), used to compute the wavefunction (in fact the part outside the
nucleus), depends on an energy parameter E occurring in its parameters a, µ, p, q. But the value of the
energy E is the energy of solution for a point-like nucleus case. This is not really suitable in the case of
the solution for a nucleus of finite size with a corrected potential.
! Problem #3. It is difficult to correctly evaluate the relativistic coefficient
γ
and the energy
parameters, such as the kinetic energy, required for a better understanding of EDOs and possible
interaction of deep-orbit electrons with nuclear fields.!
!
Finally, it should be noted that there was a rather serious problem, in our early work on EDO solutions
of the Dirac equation, about the Heisenberg uncertainty relationship (HUR), apparently unsolved also in
similar works existing in the literature: how can electrons confined in EDOs, very close to the nucleus,
respect the HUR? This issue has been solved in an unexpected way, through the use of Relativity, and a
6
change in strategy towards HUR. Therefore, it is not indicated in the previous list and the solution,
explained in detail in recent works [7], is briefly recalled in the next section.
!
2.$Semi-classical computations/simulations$
To better know the energy parameters of EDOs and their possible existence, we made semi-classical
studies [6][7][8] with many computations, by applying the following principles:
- We consider a radial potential energy PE built as a sum of inverse power terms, including the
Coulomb potential, magnetic interactions, and possibly radiative corrections
- We look for a local minimum of energy (LME) for an electron near the nucleus, while we
consider its total energy TE including the potential energy PE. TE is specified below.
- But, most important for stable orbits, the HUR must be respected.
2.1. Special Relativity and the HUR.
!
2.1.1. Respecting the HUR.
We decided to take HUR as a starting point, while considering an electron confined near the nucleus.
We previously showed [2] that Special Relativity is necessary to have deep orbit solutions of quantum
equations. Nevertheless, while starting from HUR [6][7], we find that the relativistic coefficient!
γ
!!
expected for EDOs had been greatly underestimated. In fact, electrons confined in deep orbits (EDO),
with mean radius of order a few F, are highly relativistic (
γ
can be >100). Here we summarily give some
elements and results, specified in detail in the quoted references.
For an electron confined in a sphere of radius r, from momentum |p| ~ ħ/r (to respect the HUR), and!
from its relativistic!expression p =
γ
mv, we can deduce the following results:!
!!!!!!!!!!******************
γ
!=!(1- v2/c2)-(1/2) ~ [1 + (
λ
c)2/r2]1/2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(6)!!
where
λ
c!denotes the "reduced" Compton wavelength, i.e.!
λ
c*= ħ/mc. For electrons, one has!
λ
c!~ 386 F.!
For EDOs, as it seems that r <<!
λ
c ,
γ
can be simplified into
γ
~!
λ
c/r.!We gave examples [8] showing
the expression (6) gives realistic and rather precise values.
2.1.2. Relativistic correction of the Coulomb potential energy and confinement of electron in deep orbit.
Because of the high level of the relativistic coefficient
γ
, it is interesting to consider the effects of the
relativistic correction to the static Coulomb potential, as indicated in [17][18], under the resulting form
of an effective dynamical potential noted Veff, and already considered in [2][16]. The general form (3) of
Veff, comes from the development of relativistic quantum equations (Dirac, Klein-Gordon) with the
expression of the relativistic energy of a particle in a central field for a Coulomb potential energy V*:!
! ! ! ! Veff = V (E/mc2) - V2/2mc2 (7)!
!
On the other hand, by replacing E by an approximate value
γ
mc2 in (7), we obtain the following form
(8) including the coefficient
γ
:
! ! ! ! Veff =
γ
V-V2/2mc2!!!!!!!!!!!!!! (8)
!
While putting the full expression (6) of
γ
into (8), we obtain Veff as a function of r, where
α
is the
coupling constant:
! ! ! !Veff!!= - (
α
cħ/r) ([1+(
λ
c)2/r2](1/2)!+
αλ
c/2r)! (9)
!
Finally, if r is of order a few F, one can obtain the following approximate form:
! ! ! !Veff ~!!
γ
V (10)* *
**********
So, we have the two following results:
-1. Veff is always attractive
!-2. |Veff| |V|, i.e. Veff is always a strengthening over the static Coulomb potential!energy.!!
7
Moreover, and most importantly for the EDO’s, we showed in previous works [7][8] that Veff can easily
confine an electron near the electron. Indeed, for r "very small", the kinetic energy KE = (
γ
-1)mc2 has
behavior in 1/r, whereas Veff has behavior in 1/r2. More precisely [8, fig1] we showed that for r < 2.8 F,
one has |Veff| > KE.
In fact, Relativity involvement, as defined by the HUR constraint on KE, is the solution for EDO
confinement.
!
2.2. Semi-classical computations of EDOs
In order to find EDOs, we look for a Local Minimum of Energy (LME) of electron in a central potential
PE, obtained by a balanced combination of electro-magnetic (EM) potentials near the nucleus. The total
energy is equal to TE = EH + PE, where EH is defined by the following expression:
𝐸!=!!!
!!+𝑚!𝑐!!!!!!!!!! ! (11)
It is obtained from the relativistic expression of energy of a free electron, (p2c2+m2c4)1/2, by putting |p| ~
ħ/r for respecting the HUR. Note that KE = EH - mc2.
In previous work [7], we built a combination of EM potential energies, inspired by a study [6]!of the
Vigier-Barut model [19,20,21,22,23,24] and related works, then we took radiative corrections into
account, i.e. essentially the Lamb shift (LS) [25,26,27,28,29], including two phenomena: electron Self-
Energy (SE) and Vacuum Polarization (VP). In fact, the LS supplies a specific extra energy,
corresponding to a decrease of the binding energy.! So it has a global repulsive effect on the bound
electron.
We made numerous computations of LME with variants on the combinations of EM potentials. Here,
we give only a variant of example from [8], where the LS is expressed as a repulsive quasi-potential
energy by means of extrapolations from known data tables on QED effects on orbital parameters. In
fact, we put VLS = 0.623x10-10/
ρ
3 in J/F3. In Fig.2, we plot an example curve of the binding energy BE =
KE + PE = TE - mc2, where the LME!corresponds to!
ρ
~ 1.4 F!
!
!
!
In this example, the main parameters of the electron at the LME (at
ρ
~ 1.4 F) have the following
values:
γ
~ 275
BE ~ - 509 keV
• PE ~ - 140.5 MeV
• KE ~ 140 MeV
2.3. Key information from semi-classical calculations.
Fig.*2. Plot of electron BE,
for 1.3 F<!
ρ
< 1.55 F
!
V,*keV!
ρ
,F
!
8
Semi-classical studies provided a lot of useful information for correcting the imperfections of the initial
Dirac EDO’s for finite nucleus case. Not only we can evaluate the relativistic coefficient γ, but also all
the energy parameters. Moreover we found that the relativistic virial theorem [30] was respected by
ultra-relativist EDOs, in a very simple form and for various combinations of potentials. So, we can see
that
• Electrons confined near the nucleus are ultra-relativistic
• The relativistic virial theorem is respected at the LME, in the following form:
KE/ |PE| ~
γ
/(
γ
+1)
This allows us to deduce all the main parameters of the Dirac EDOs, computed at the mean radius of the
probability density distribution, and to check their coherence: With a corresponding LME radius -->
mean radius <r> of Dirac EDO solutions with finite nucleus, we can evaluate
γ
, TE, KE, and BE at
<r>. For example, one can show TE ~ mec2/
γ
~ mec2 <r>/
λ
c .
So, we can correctly adjust the energy levels of Dirac EDO's solution (see Problem #2), and know all
the energy parameters (see Problem #3).
3. How apply the new information provided by semi-classical computations, to Dirac EDOs?
3.1. A brief reminder of what the solution of a quantum equation for wavefunction consists of.
Such a quantum equation generally has the following form: H
ψ
= E
ψ.
Summarily, we have two concomitant mathematical issues.
1. H is a "Hamiltonian", representing the total energy of the considered system. It is generally a
Hermitian operator including differential operators. For example, the momentum p is expressed by the
expression 𝑖 !
!" , including a vector of partial derivatives on spatial coordinates, !
!" .
Note a Hamiltonian can be multidimensional (e.g. in the Dirac equation). So, we have to solve a
differential equation.
2. We look for
ψ
, an unknown wavefunction, and for E, unknown eigenvalues, which are energy
values associated with eigenvectors
ψ
corresponding to wavefunction solutions. These can be
characterized by integers, "quantum numbers", if the set of eigenvalues is a discrete set, composing
energy levels.
So, we have to solve a "spectral" problem [31] for H atom: to find eigenvalues associated with solutions
of the differential equation.
3.2. Solving the Dirac equation for hydrogen atom with finite nucleus.
3.2.1. Solving the system of radial differential equations.
After separation of the angular and radial variables, one ends up with a system of two radial equations.
As we consider a H atom with non-point-like nucleus of radius ~ R, we have to solve radial differential
equations, where the radius belongs to two separate domains of the real numbers, associated with the
inside and the outside of the nucleus respectively. And the central potentials are very different in these
domains.
(i) Outside the nucleus, the static potential energy corresponds to the classical Coulomb potential, and is
expressed by means of the formula VO = -e2/r, where e is the charge of the electron (expressed in
suitable units)
(ii) Inside the nucleus, the potential energy VI is expressed by the chosen formula (5), with Z = 1,
equivalent to VI = - (e2/2R) (3-r2/R2)
So the whole potential energy is described by a piecewise expression:
Pot(r) = VI, if 0 r R , and = VO, if r > R.
As a consequence, the differential equation Eq(r), using Pot(r), will have a piecewise form:
Eq(r, Pot(r)) = EqI (VI), if 0 r R, and = EqO (VO), if r > R.
9
The software used for differential equations, Maple, can solve any differential equation on a limited
domain. For the system of Dirac radial equations, we can observe the following:
1. First, the formal solutions are very different in the two domains. As complex expressions
including very different special functions (defined by series), they are very difficult to unify.
2. A solution on the limited domain considered (here the domain outside the nucleus) is formally
the same as on the whole domain of real numbers.
Note we apply this reasoning only on the limited domain outside the nucleus: for the very small domain
inside the nucleus, it is sufficient to solve the differential equations approximated by polynomials.
From the point #2, we deduce the general solution for point-like nucleus case, i.e. the solution on the
whole domain of real numbers, can be used for the domain associated with the nucleus outside. So, for
this domain, we can use, in principle, the solution indicated in section 1.2, i.e. the expression (4) of the
pairs of functions (f, g). Nevertheless, as noted in Problem #2, section 1.2.3, the expression (4) used to
compute the wavefunctions solutions of the radial equations, depends on an energy parameter E
occurring in the expressions of several parameters, a, µ, p, q. But the only value of E that we know is
the one given by formula (3), corresponding to the energy levels of Dirac solutions for point-like
nucleus case. Of course this is not really suitable in the case of solution for a nucleus of finite size with
a corrected potential. When applying the general solution to limited domain r > R, for solutions outside
the nucleus, the eigenvalue E’ associated with the wavefunctions (fO,gO) must be different from the
energy E of general solutions for point-like nucleus. So, we have to solve a spectral problem.
To address this issue, we use a kind of perturbative method, in the form of iterative computation until
we reach a fixed point. It is explained in subsection 3.2.3.
3.2.2. Connecting inside and outside solutions (Problem #1)
Before the fixed-point process, we consider the question of connecting the couples of inside and outside
solutions at the nucleus surface, with continuity of the total functions and their derivatives.
More precisely, the couples of inside (fI,gI) and outside (fO,gO) solutions have to be connected at the
“surface” of the nucleus, under the following conditions (Problem #1, section 1.2.3):
(i). We must satisfy the continuity at r = R , i.e. fI (R) = fO(R) and gI(R) = gO(R)
(ii) We must satisfy the continuity of respective derivatives
fI’|R = fO’|R and gI’|R = gO’|R
Here we indicate, in a rather brief and simplified way, the successive steps that allow us to satisfy these
conditions.
The functions fO and gO determined by the expression (4), include the same multiplicative constant C,
because of the coupling of the radial equations: i.e. we have fO = C Expf, gO = C Expg, where Expf and
Expg are expressions deduced from the computation of (4) and represent two functions of the radius r.
I. As a first step, we decouple these two functions, while writing fO = C1 Expf and gO = C2
Expg.
II(a). While using an ansatz as in section 1.2.1, we compute a polynomial Pf as an approximation
of fI. So fI is expressed in powers of the radius r and includes the free parameter
λ
f.
We compute also the derivative fI' in r of fI; fI' also includes the free parameter
λ
f.
(b). Then, we compute the derivative fO' in r of the function fO. Of course, the function fO'
includes the multiplicative parameter C1.
(c). Finally, we calculate fI(R), fO(R), fI'(R) and fO'(R), where R is the "junction" radius between
the inside and outside of the nucleus. Then we solve the system of equations {fI(R) = fO(R), fI'(R) =
fO'(R)}. The solution of this equation gives a result in the form {C1 = a,
λ
f = b}, where a and b are two
real numbers, and we replace C1 and
λ
f by their values in the expressions for fI, fO, fI', fO'.
III. A similar process is applied to the second component, represented by the couple of functions
(gI, gO), which leads, after solving a system of equation at the junction radius R, to a result having the
form {C2 = c,
λ
g = d}, with two real numbers c and d.
10
IV. In order to restore the initial coupling of components f and g, we put FI = c fI, FO = c fO and
similarly for their derivatives (just for checking the condition (ii) above).
For the g components, we multiply by a, i.e. GI = a gI, GO = a gO,...
Finally, we denote, by F = fIUfO, the total function obtained by connecting fI with fO at r =R, and
similarly G = gIUgO.
3.2.3. Fixed Point method for solving the spectral problem (Problem #2).
We indicate in very simplified form, the iterative process used.
Starting point: for a given value of radial quantum number n', the expression (3) of energy for solution
in the point-like nucleus case, gives a value noted E0.
• Step #0: we take E0 to determine the total wavefunction
Ψ
0 = (F, G), as explained in the prior
sub-section, and we compute the electron probability density. Then we deduce the average orbital radius
r0 and we can directly calculate the total energy E1 of electron at r0, as explained in (2.3). Next we will
go to step one, where a new value r1 of orbital radius is computed.
Step #1: Normally we have E1 E0. Then, as in Step 0, E1 determines a new wavefunction
Ψ
1
and we compute the mean orbital radius r1 from the new electron probability density distribution. From
this, we calculate the new energy E2 at the radius r1
(i) If E2 ~ E1 (up to 3 digits), we consider that we reached a fixed point
(ii) If E2 E1, we go to a Step #2 similar to Step #1.
In fact, many computations, for different values of the radial number n' and even by varying the "initial"
value of energy E0, lead to the following observations:
- The mean radius corresponding to wavefunction
Ψ
varies very slowly as a function of the
energy E injected in expression (4) to calculate
Ψ
.
- At step #1, r1 is already close to r0.
- We can stop the process at the step #1 to check that the fixed point is practically reached and, if
so, take
Ψ
1 as a wavefunction solution of the problem and E1 as total energy of this solution.
3.2.4. EDO Solutions of Dirac equation for H atom with finite nucleus.
In Fig.3, we plot the curves of normalized probability density corresponding to n'=1 and n'=2, while
considering a junction radius R = 1F, and for 0.4 F <
ρ
< 3, where
ρ
denotes the radius in F.
Fig.3. Plot of the normalized
probability density, for n'=1 (red)
and n'=2 (blue)
We give the mean radius <
ρ
>, and the values of the relativistic coefficient
γ
and the kinetic energy KE,
which were not obtained with the initial results indicated in section 1. Moreover, we compare the values
in F
11
for the total energy, denoted by TE, and the binding energy BE obtained with our new method, with
those obtained in the initial solutions, corresponding to the point-like nucleus case (PLN), written in
italic.
n'=1, <
ρ
> ~ 4.5 F,
γ
~ 84,
KE ~ 42.5 Mev, TE ~ 6 kev (1.8 kev, PLN), BE ~ -505 kev (~ 509 kev, PLN)
n'=2, <
ρ
> ~ 1.13 F,
γ
~ 405,
KE ~ 206 Mev, TE ~ 1.5 kev (1 kev, PLN), BE ~ -509.5 kev (~ 510 kev, PLN)
Moreover, one can verify the mean radius <
ρ
> and the total energy TE satisfy the following
relationship: <
ρ
> ~
λ
c TE/mc2, where
λ
c is the reduced Compton wavelength (~ 386 F) and m is the
electron mass.
4. Can EDO states be populated?
4.1. Heisenberg barrier
Here is a first attempt to evaluate a possible population of EDO states by tunneling from the atomic-
electron ground state, in the form of superposition of quantum states. To simplify the situation, we
consider only two antagonistic interactions, one due to the attractive relativistic effective potential
energy Veff, and the other to the Heisenberg uncertainty relationship (HUR), increasing the kinetic
energy to prevent the containment of electron: we call it “Heisenberg barrier” [8](p!478).
These are not static fields, but dynamic effects associated with a possible increasing confinement of the
electron.
• On one hand, we consider the kinetic energy KE = (
γ
-1) mc2
• On the other hand, the relativistic effective potential energy Veff =
γ
V-V2/2mc2,
where the relativistic coefficient
γ
for an electron confined at a radius
ρ
is given by
γ
~ [1+ (
λ
c/r)2]1/2
In Fig. 4, we plot KE and |Veff|, for 1 F <
ρ
< 100 pm, to clearly show the existence of three zones:
Fig.4. Loglogplot of KE and |Veff|,
for 1 F <
ρ
< 100 pm
1. For 26.5 pm <
ρ
, the
"atomic" zone, where KE < |Veff|
2. For 2.8 F <
ρ
26.5 pm,
a zone where KE > |Veff|
3. For 1 F <
ρ
2.8 F, an EDO zone, where KE < |Veff|
Remark. In fact, the presence of EDO electrons is not limited to the zone #3, since we found an EDO
wavefunction having a probability density with mean radius at 4.5 F, because of the very large tail
extending far beyond the range displayed in Fig. 3. But here, we only considered the strongly attractive
potential Veff, in a simplified semi-classical analysis.
12
We make the assumption that an increasing containment of electron wavepacket is possible.
The start of this increasing containment is possible, because in the atomic zone #1, we have KE < |Veff|,
which can push a (tiny) fraction of the electron wavepacket, whose maximum probability of presence is
at the Bohr radius, i.e. at 53 pm ~ 2 x 26.5 pm, to move towards the nucleus until 26.5 pm.
But, arrived at 26.5 pm and for reaching the zone #3, the wavepacket should have to go through the
zone #2, where the repulsive effect of kinetic agitation required to satisfy the HUR is greater than the
attractive effect of the potential Veff.
We say the zone #2 forms a "Heisenberg barrier" between the zone #1 and the zone #3. Of course, this
“Heisenberg barrier” is a virtual dynamic barrier, unlike the usual potential barriers. In addition, there is
a reversal of roles between potential energy and kinetic energy: in the usual cases, the kinetic energy
KE pushes an electron to cross a barrier, while the potential energy Pr is repulsive and prevents the
electron from crossing the potential barrier, and if Pr > KE, a pure classical mechanical reasoning
prohibits the crossing. Then only a quantum process of tunneling allows a fraction of the amplitude of
the wave associated with the electron to cross the barrier.
Here, it is the attractive dynamic potential of Veff,, that tends to push the electron to cross the barrier for
having an increasing containment, whereas the increasing kinetic energy KE due to the containment, a
reactive agitation implied by the HUR, tends to push the electron back. And in pure classical
mechanical reasoning, if |Veff| < KE, the containment of the electron should not increase. But we
consider an extension of the quantum tunneling process to this specific situation, allowing a fraction of
the amplitude of the electron wave to cross the Heisenberg barrier to reach zone #3.
If considering ΔE = KE -|Veff|, we can find a maximum of ΔE, equal to ~17 MeV, at
ρ
~ 5.6 F. This is
the point at which the barrier to orbital stability is greatest. We plot the curve of ΔE in the figure 5, for
2.8 F <
ρ
< 27 pm.
Fig.5. Semilogplot of ΔE for 2.8 F<
ρ
<27 pm
4.2. Tunneling through the Heisenberg barrier
To compute a possible tunneling, we use the WKB (Wenzel-Kramers-Brillouin) approximation [32]
[33] in one dimension, similar to the calculation of the Gamow astrophysical factor [34].
We put K(
ρ
) = [2m ΔE )]1/2, where ΔE = KE-|Veff|. Note that K(
ρ
) has physical dimension of momentum.
We plot the curve of K(
ρ
)x1021, for 2.8 F <
ρ
< 27 pm in Figure 6 (the multiplicative factor 1021 is only
used to simplify the writing of values on the vertical axis).
We can remark the similarity of the shape of curves in fig. 5 and fig. 6.
ρ in pm
MeV
13
Fig.6. Semilogplot of K(
ρ
)x1021for 2.8 F<
ρ
<27 pm
Then, we compute 𝑄 = 𝐾(𝜌) 𝑑𝜌
!!
!!, where
ρ
0 ~ 2.8 F,
ρ
1~ 27 pm, and we put wexp = Q /ħ = 5.65
which is a dimensionless number.
Note that the integral Q of K(
ρ
) corresponds to the blue area (divided by 1021) below the curve of K(
ρ
).
The weakening factor of the electron wave amplitude is given by w ~ e-wexp ~ 0.003.
So, the electron presence probability is P = w2 ~ 9 x 10-6.
We interpret this result by saying that the general wavefunction of an 1S electron orbital could be a
linear combination of EDO state and atomic ground state, with amplitude coefficient
λ
EDO ~ 0.003.
Remark. In [35], the authors also consider that a general solution of the Dirac equation for an electron
(or a muon) bound in an atom is a linear combination of the regular (usual) solution and an "extra"
solution, usually neglected; Indeed, this latter solution is eliminated when considering a point-like
nucleus; but, it becomes acceptable when taking into account the size of the nucleus, which eliminates
the singularity at the origin. Nevertheless, in their numerical results on many atoms, they do not provide
the coefficients of linear combinations but the energy shifts due to the size of the nucleus, which is the
purpose of their work.
Conclusion and prospect
1. Semi-classical simulations carried out in our previous works, allow us to solve questions about EDOs
as solutions of the Dirac equation with finite nucleus. Indeed, while making semi-classical analysis and
computation, we showed the following results:
(i). For an electron confined near the nucleus, the HUR requires a strong containment energy but
allows us to determine a value for the relativistic
γ
(ii). We know that the EDO solutions are highly relativistic, which is an important result for LENR,
and as a consequence, the magnitude order of KE can be in the 100 MeV range
(iii). The effective potential, Veff, a relativistic correction of the Coulomb potential deduced from
relativistic quantum equations (Dirac or Klein-Gordon equations), can confine deep-orbit electrons
within the nuclear region."]
(iv). For a deep electron having a local minimum of energy (LME) and respecting the HUR, we can
compute the relativistic coefficient
γ
, the total energy TE, the kinetic energy KE and the binding energy
BE, at the radius r of the LME.
While going back to EDO solutions of Dirac equation for atomic H, by considering a finite nucleus:
(i) We can recalculate the radial wavefunctions of Dirac EDO solutions and determine the correct
energy level E associated with the radial quantum number n’, and moreover we can compute the
14
coefficient
γ
, TE, KE and BE for electron at the mean radius <r> of the probability density distribution.
So, we solve problems #2 and #3 (subsection 1.2.3, Obtained results and discussion on imperfections)
(ii) Moreover, we obtain wavefunctions with smooth continuous shapes and having continuous
derivatives everywhere i.e., we solve problem #1.
2.1. Nevertheless, the Dirac equation seems to provide an EDO solution for each value n' of the radial
quantum number, whereas semi-classical computations that do not include resonances between
electron-orbital and emitted-photon frequencies as described in [36], give only a single EDO solution.
On the one hand, this behavior difference between the quantum equation and semi-classical results also
happens for "regular" solutions: semi-classical computation of electron LME in H atom, carried out
while considering only the total energy, provides only the ground (Bohr) state, while a quantum
equation, e.g. the Schrödinger equation provides all the excited states, corresponding to increasing
values of the main quantum number n (involving the angular quantum number l, as n = n' + l +1).
On the other hand, unlike the regular solutions while considering Dirac EDO wavefunctions, the mean
<r> radius of the probability radial density distribution decreases when the radial number n' increases.
As a consequence, and since <r> decreases on a bounded interval, the sequence <r>(n') of the values of
<r> as function of n', has an accumulation point when n' tends towards infinity. In other words, it means
that for high values of n', the wavefunction solutions are practically and physically indistinguishable.
The meaning of this situation would require further physical interpretation. Can we say that because of
quantum fluctuations, from a certain integer N to be defined, all the solutions for n' > N are confounded,
and that finally there is only a finite number N of solutions?
2.2. Another question about the Dirac solutions arises, when comparing the characteristic energy
parameters of these solutions and those of the semi-classical computations: it seems that semi-classical
solution, with LME at radius r ~ 1.4 F, is closer to the Dirac solutions for n' = 2, where <r> ~ 1.13 F
than the one for n' = 1, where <r> ~ 4.5 F. Of course, we know the Dirac results are dependent on a
somewhat arbitrary choice of the junction radius R between inside and outside the nucleus. But even
with "reasonable" decrease of R, we still have a significant gap between the semi-classical solution and
the Dirac solution for n' = 1: for example, by putting R = 0.84 F, i.e. < 0.87 F, which is at the present
time the "official" value of charge radius of the proton, we obtain a mean radius <r> ~ 3.95 F for n' = 1.
And on another hand, despite many semi-classical calculations with various combinations of potentials,
we never found a LME at a radius r > 2 F. The question therefore arises as to what the solution for n' =1
physically represents, if it is not the "basic" EDO state.
3. At section 4., we only began to address the issue of EDO population, by using the WKB
approximation for tunneling from atomic state to EDO state. On one hand, the WKB method has been
applied in dimension one, when it would make more sense to do it in dim 3. For the time being, various
attempts in this direction have not led to realistic results, and the question remains to be addressed.
On another hand, a possible lead would be to look for physical parameters in condensed matter, likely
to increase the tunneling toward EDO states.
Finally, it should be noted that the analysis carried out in subsection 4.1 allows us to give an answer to a
legitimate question posed by the referee: "why all hydrogen atoms in the Universe (and this is about
74% of all atoms in the Universe) do not spontaneously transfer from “standard” states to these
superdeep levels?".
Indeed, we can reasonably expect that the existence of the "Heisenberg barrier" prevents the electron
from spontaneously transferring from standard states to superdeep levels.
This also answers the naïve question encountered in forums: "why the electron doesn't fall into the
nucleus?"
4. Finally, LENR features such as energy transfer with neither gamma radiation nor energetic particles,
requires enhanced internal conversion. So, we study possible connections between highly energetic
deep electrons and nuclei, hadrons and quarks.
15
Acknowledgement
This work is supported in part by HiPi Consulting, Windsor, VA, USA; and by the Science for
Humanity Trust, Inc, Tucker, GA, USA.!!
The authors would like to thank the reviewer for his judicious advice and recommandations allowing us
to improve the comprehension of our work.!
!
!
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DOI:https://doi.org/10.1103/PhysRev.72.339
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Since the discovery of the corpuscular nature of radiation by Planck more than fifty years ago the quantum theory of radiation has gone through many stages of development which seemed to alternate between spectacular success and hopeless frustration. The most recent phase started in 1947 with the discovery of the electromagnetic level shifts and the realization that the exist­ ing theory, when properly interpreted, was perfectly adequate to explain these effects to an apparently unlimited degree of accuracy. This phase has now reached a certain conclusion: for the first time in the checkered history of this field of research it has become possible to give a unified and consistent presen­ tation of radiation theory in full conformity with the principles of relativity and quantum mechanics. To this task the present book is devoted. The plan for a book of this type was conceived during the year 1951 while the first-named author (J. M. J. ) held a Fulbright research scholarship at Cambridge University. During this year of freedom from teaching and other duties he had the opportunity of conferring with physicists in many different countries on the recent developments in radiation theory. The comments seemed to be almost unanimous that a book on quantum electrodynamics at the present time would be of inestimable value to physicists in many parts of the world. However, it was not until the spring of 1952 that work on the book began in earnest.
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An explicit non-relativistic mathematical analysis of a model proposed by Vigier to interpret (within the present frame of quantum theory, i.e. in terms of spin-orbit and magnetic interactions appearing in dense media) excess heat observed in the so-called “cold fusion” phenomena based only on hydrogen is presented. The existence of new “tight” Bohr orbits is demonstrated in this case.