2
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Centro Studi di Radioattivit
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E  mail: conte@teseo.it
A SUGGESTED EXPERIMENT OF RADIATION PHYSICS TO ANALYZE
A NEW KIND OF FUSION FOR NUCLEAR PARTICLES.
E. Conte, M. Pieralice (+)
(+) C.N.R.  Bari  Italia
Lavoro eseguito con il contributo del C.N.R.
Note that this diskette or file is an integrating part of the book "Meccanica Q
uantistica Biquaternionica" (title:Biquaternion Quantum Mechanics; author:Elio
Conte; Publishers: Pitagora Editrice, Bologna Italia).
Please, send your order of the book to the E  mail: conte@teseo.it
INTRODUCTION
This paper is devoted to the announcement of a new process of fusion f
or nuclear particles.
Starting in the 1990's (Conte 1993, 1994 a, b, c), we have developed a generali
zation of quantum mechanics using the biquaternions. We have called this new th
eory the Biquaternion Quantum Mechanics. This theory (Conte 1993, 1994 a, b, c)
solves some basic problems that had remained unsolved in the basic formulation
of quantum mechanics in 1927 by the School of Copenaghen (D'Espagnat, 1977); i
n addition, this new theory enables us to generalize the usual Schrdinger equa
tion (Conte, 1995).
In this paper we will discuss this equation showing that a new process of fusio
n is possible on the basis of this equation.
A BRIEF NOTE ON THE BIQUATERNIONS
In our previous papers (Conte, 1993, 1994 ac, 1995), we introduced an
accurate exposition of the algebra of the biquaternions. Our present discussio
n will be devoted to the reader unfamiliar with this kind of hypercomplex numbe
rs.
Consider the following set of basic elements (1, e1 , e2 , e3 ) where 1 is the
well known scalar unity, and ei (i = 1, 2, 3) are three anticommuting elements
ei ej =  ej ei ; i = 1, 2, 3 ; J = 1, 2, 3 ;
i # j
ei2 = 1 ; ei # 1
(2.1)
The set (1, e1 , e2 , e3 ) represents the four generators of our algebra, and,
in matrix form, ei are expressed by the well known Pauli's matrices

@
0
(
@
0
Let us give now a proper definition of biquaternion. It is a quaternion having
complex components
Z = z0 + z1e1 + z2e2 + z3e3
(2.3)
where zm (m= 0, 1, 2, 3) are complex numbers. The hyperconjugate Z* of Z is the
biquaternion
Z* = z0  z1e1  z2e2  z3e3
(2.4)
while the conjugate Z+ of Z is the biquaternion
Z+ = z*0 + z*1e1 + z*2e2 + z*3e3
(2.5)
The norm of Z is
N(Z) = Z Z* = Z*Z = z20  z21  z22  z23
(2.6)
The biquaternion Z1, the inverse of Z, is
Z1 = [N(Z)]1Z*
(2.7)
with N(Z) # 0.
The relations from the (2.1) to the (2.7) complete our brief exposition on the
biquaternions.
Let us generalize, now, the quantum mechanics by the biquaternions.
As we have shown in (Conte, 1994c), the biquaternion quantum mechanics uses Lin
ear Homogeneous Biquaternion Transformations, LHBT (.,.,.), to generalize the s
tatements of the usual quantum mechanics. Given the biquaternion Z, and the biq
uaternions I = U U+ = U+ U, and A with N(I) = 1 and N(A) = 1, we have that
LHBT(I = U U+, Z, A) <====> Z' = U Z U+A (2.8)
Particular cases of the (2.8) are given as it follows
LHBT(I = U U+, Z, 1) <====> Z'= U Z U+ (2.9)
or
LHBT(1, Z, A) <====> Z'= Z A ; A = U U+ (2.10
)
U is a given biquaternion.
As we know, the first stage of the usual quantum mechanics involves the notion
of commuting or noncommuting observables that are characterized by the notion
of commutator between linear and hermitean operators A and B. We have
[A, B] = A B  B A
(2.11)
By using the (2.9), the (2.11) may be generalized in the following manner. Cons
ider U to be a biquaternion with U U+= U+U # 1. Thus, we indicate U U+= U+U = I
. We may calculate that
U [A, B ] U+ = U (A B  B A) U+= U A B U+ U B A U+= U A U+(U U+)1 U B U+
 U B U+(U U+)1U A U+= A'Q B' B'Q A'= [A', B']
(2.12)
where we have posed
A'= U A U+ ; B'= U B U+ ; Q = (U U+)1
We can see that (2.12) is a generalization of the theory. For a detailed exposi
tion of principles and rules of biquaternion quantum mechanics see (Conte 1993,
1994 ac).
ON A GENERALIZATION OF SCHRDINGER'S EQUATION USING BIQUATERNIONS
Consider the case of (2.11) in which A is the operator for the positio
n of the usual quantum mechanics, A = x, and B the operator for the momentum,
B = ihd/dx, (h=h/2p). In (2.11) we have the commutation relation regarding the
celebrated Heisenberg's uncertainty principle
[ A, B ] = i h
(3.1)
Owing to the (2.12), we have that
[ A', B' ] = i h U U+= i h I
(3.2)
This is, precisely, the important result that we must consider. The way to gene
ralize Schrdinger's equation by biquaternions implies necessarily to consider
that the new quantization, instead of Planck's constant, becomes a dynamical va
riable in the induced quantum theory. As consequence,this new theory, generaliz
ed with respect to the usual quantum mechanics, is finalized to analyze physica
l systems where it is of importance to examine the consequences of quantum fluc
tuations I in the quantum of action h.
This result represents the essential landmark that animates this paper. Our sta
rting point is that we have to consider the following modification
h h I (r,t)
(3.3)
I (r,t) is the biquaternion U U+, and, for brevity, we will consider here only
its real part.
We may generalize, now, Schrdinger's equation.
As it is well known, considering the canonical action (Janussis, 1978) A = ihl
ogy, where y = y(r, t) is the state of the system in the conventional Hilbert s
pace, we have that
dA/dt + H = 0 ; dA/dxk= pak
(3.4)
that may be rewritten in the familiar form of the fundamental Schrdinger's equ
ation of quantum mechanics
i h dy/dt = H y
(3.5)
with  ih "ky = pky . This is easily obtained inserting A =  ihlogy in the (3.
4).
This is the way to obtain the usual Schrdinger's equation.
In our case we have to use the (3.3) and, thus, the new action A that will be g
iven as it follows A =  ihIlogy.
We may consider that it must be
(dA/dt) y = H y
(3.6)
with
y(r, t) = y0exp( kt)
(3.7)
Considering the (3.3), we have that
(i h dI/dt logy  i h I k) y = H y
(3.8)
Let us examine the simple steadystate case for which dI/dt = 0. From the (3.8)
we obtain that
H Q y = E y
(3.9)
where Q = I1, and K = i E / h.
In the way of a generalization of Schrdinger's equation, we have obtained (3.9
) which, however, does not represent a generalized Schrdinger's equation.
In the usual derivation of Schrdinger's equation we consider that
1 / v(x, y, z) = f(n) [ E(n)  V(x, y, z) ]
where V(x, y, z) is the velocity of phase for the wavepacket. It results that
fn = a = costant
Planck's constant is usually introduced in the following manner
(2 m) / a = h
If we consider a fluctuation h I(r,t) of h, we have to admit consequently that
m and a are modified. Thus we have in particular that m' = m r0 where r0 is a p
arameter to be determined (r0# 1). We will discuss in detail this renormalizati
on of the mass m in the following section.
Let us return, now, to consider (3.9). There is a case in which (3.9) still may
represent a generalization of Schrdinger's equation. Consider a quantum syste
m where the potential V(r) acts. The usual Schrdinger's equation is
H m = Em m
(3.10)
Let us consider that
Q(r) = I1(r) = 1   m >< m 
(3.11)
We may insert (3.11) into (3.9), and we obtain
H (1   m >< m) y = E y
(3.12)
that may be rewritten in the following terms
H ' y = E y
(3.13)
with
H ' = H  1/r0 Em < my > m/y
(3.14)
In this manner we have the (3.13) that now represents the required generalizati
on of Schrdinger's equation. Indicating by
V ' (r) =  1/r0 Em < my > m/y
(3.15)
the generalized equation (3.13) assumes the following form
[  h2/(2 m') "2 + V(r) + V '(r) ] y = E y
(3.16)
It is connected to the usual Schrdinger's equation (3.10) which is expressed i
n the following manner
[  h2/(2m) "2 + V(r) ] m = Em m
(3.17)
The purpose of this paper is now achieved. We have shown that, starting with (3
.17) describing a quantum system by the usual Schrdinger's equation, we may ge
neralize the (3.17) in the form of a new Schrdinger's equation, (3.16), so to
give the expected biquaternion generalization. We have still to define some add
itional points.
The first point is that in the present paper we have used an elementary derivat
ion of the generalized Schrdinger's equation (3.16). For a detailed discussion
, see (Conte, 1995).
The second point is that, deriving (3.16), we have used a method called the emp
irical pseudopotential transformation that is not new here . Starting with 195
9 it was used by J.C. Philips and L. Klainman (1959) for different purposes, es
sentially as a quantum technique to compute wave functions in molecules and cry
stals.
APPLICATIONS OF A GENERALIZED SCHRDINGER'S EQUATION
This section constitutes the true aim of our paper.
Let us examine the different problems that are posed at the level of applicatio
n. In applying (3.16), the first problem regards the determination of V '(r) th
at is unknown.
As the first step, we have to clear the field of phenomenology to which the equ
ation (3.16) is applied. We have two orders of considerations to this regard.
The usual Schrdinger's equation describes interacting particles at atomic leve
l and it has been employed also to describe phenomena at nuclear level but with
nonentirely satisfactory results. Thus, a generalized Schrdinger's equation
should apply to nuclear phenomena at distances shorter than 1013 cm and for pa
rticles that are not considered to be pointlike.
We may go now to examine V '(r).
In the usual quantum mechanics, the range R of an interaction depends on h, on
c, the speed of light in a vacuum, and on V0, the potential energy of the inter
action. We have that
R = f ( h, c, V0)
(4.1)
as shown by Mickens (1979).
It may be expressed in the following manner
R  V0  = h c
(4.2)
or
R  V '(R)  = h c
(4.3)
In our generalized case, the following modified relation corresponds to (4.3)
V '(R) = h I(R, t) c / R
(4.4)
In this manner V '(r) depends from V (r) of (3.17) owing to the presence of I(r
, t), being I = Q 1 with Q = 1   m >< m. Consider the case V (r) =  e2 /
r. It follows that
m = m (r) @ c exp( a r)
(4.5)
From (4.4) we have that
V '(R) @ (h c / R) 1 / [1  exp( 2aR)] V '(r) = V0 exp( 2ar) / [1  exp(2a
r)]
(4.6)
for r 0.
We have obtained the explicit expression for V '(r). It is a kind of Hulthn po
tential, well known in nuclear physics. Equation (4.4) supports the existence o
f an Hulthn potential V '(r) when V (r) is a pure Coulombian interaction.
We have confirmed that the (3.15) is an Hulthn potential in the general formul
ation of (3.16) by using numerical methods and V (r) =  e2/ r.
Also the analytical solution of the (3.16) confirms the (4.6).
We are now in the condition to analyze (3.16) in the case in which V(r) is a pu
re Coulombian interaction.
Equation (3.16) is rewritten as follows
[ h 2/ (2 m') "2  e2/r  V0elr/ (1  elr)] y = E y
(4.7)
where
V (r) =  e2/r @  e2l elr/ (1  elr) for r 0
(4.8)
On this basis, (4.7) may be rewritten as follows
[  h2 / (2 m') "2  V0' elr/ (1  elr)] y = E y
(4.9)
with V0'= e2l + V0.
As we know, Hulthn potential usually is an attractive interaction with a const
ant value of relatively high value; in consequence in (4.9) we may consider a c
umulative attractive effect where the dominant role is played by the Hulthn i
nteraction.
Equation (4.9) may be solved for l = 0 in the standard radial form as it was ob
tained by Flugge (1971). introducing the dimensionless variable x = l r, we hav
e the radial differential equation
d2c / d x2 + [ b2 ex / (1  ex)  a2) c = 0
(4.10)
where
b2 = (2 m'V0') / (h2l2) ; a2 = ( 2 m'E) / (h2l2) ; c(r) = r y(r
) (4.11)
with a > 0 and b > 0.
We have reached the crucial stage of our formulation. The solutions of (4.10) a
re wellknown (Flugge, 1971). The important conclusion is that (4.10) admits bo
und states. These are bound states that imply the penetration of the considered
particles and, thus, imply their fusion.
Note that it must be
a = (b2  n2) / (2 n)
(4.12)
and a > 0, with n = 1, 2, 3, .... . The Binding energy is given in the followin
g manner
En =  V0'[ (b2  n2) / (2 n b) ]2
(4.13)
and, for n = 1, we have that
E1=  (h2l2) / (8 m') [ (2m'V0') / (h2l2)  1 ]2
(4.14)
For details, see Flugge (1971).
We have reached a conclusion that may be of the greatest importance.
Our generalization of Schrdinger's equation admits the possibility of the fusi
on for the nuclear particles, and this result could open new frontiers of scien
tific knowledge.
A PROPOSED EXPERIMENTAL TEST
Our aim is to evaluate (4.14) for specific cases of interest, and to p
ropose an experimental verification of these cases. Here, we will discuss the m
ost simple system that may be explored on the basis of (3.6) and (3.17). Consid
er a system constituted by a proton and an electron. We will call it the system
(p, e) for the fusion.
For simplicity, consider the proton at rest.
Let us evaluate the renormalization m' = m r0.
We have shown (Conte, 1995a) that the use of (2.9) changes our well known relat
ivistic invariant
r2 = x0x0  x1x1  x2x2  x3x3
(5.1)
that regards only the empty space. If we consider a given medium that is differ
ent from empty space, a new generalized relativistic invariant must be consider
ed r'2= x0b02x0  x1b12x1  x2b22x2  x3b32x3
(5.2)
b0 = b0(......) ; b1 = b1(.......) ; b2 = b2(.......) ; b3 = b3(...
....) (5.3)
In (5.2), bm = bm(......, c, .......) are characteristic functions that depend
on the physical features of the medium taken in consideration. This result impl
yes that a particle, moving in the medium taken in consideration, needs (Conte,
1995a) a new relativistic representation of its energy as m0c2b02 instead of m
0c2 as in the usual case. This is, precisely, the case of the electron moving t
o penetrate inside the proton whose medium has specific bm characteristic funct
ions. Thus, in our experiment, the electron mass me' must be changed as follows
me' = mec2b02
(5.4)
R.M. Santilli (1990, 1994), following the indications of founder fathers of phy
sics and in particular of Fermi (1947), was able to suggest the invariant (5.2)
in 1988 without the use of the biquaternions that, instead, we applyed (Conte,
1995a) on the basis of a procedure of generalization of the physical laws that
uses (2.9). This author also estimated the value b0 = 1.60.
In conclusion, using this value, we have that the fusion (p, e) has in our mod
el the proton at rest and the electron with energy me' = 1.306 Mev.
Let us consider, now, equation (4.12). Since a > 0, for n = 1 (the ground state
of the (p, e) ), it must be that b2 > 1. In other terms, we have that
2 m' V0' / (h2l2) > 1
(5.5)
Consider R = 0.9*1013 cm, thus we have that l = 1 / R = 1.11*1013 cm1. On the
basis of (5.5), we have that
V0' > 2.770*102 erg
(5.6)
i.e., chosing V0' to be 2.782*102 erg, we have, from (4.14), the value of the
binding energy for the (p, e) system. We obtain that
E =  64 keV
(5.7)
Choosing V0' to be 2.783*102 erg, we have
E =  76.5 keV
(5.8)
In conclusion it is important to observe that for values of V0'above the thresh
old value fixed by (5.5), we may have binding energy values that are above some
keV for each fusion.
Let us consider another result of particular importance.
For (2.782*102 < V0' < 2.783*102 ) erg, we have the value E =  72 keV of the
binding energy.
Let us examine the importance of this result: our (p, e) system has the rest m
ass of the proton (938.279 MeV), and has the renormalized mass of the electron
that is 1.306 MeV, as previously calculated. Thus, the (p, e) system has the f
ollowing rest energy:
(p, e) system ; (938.279 + 1.306  0.072) MeV = 939.513 MeV
Note that this value represents the rest energy of the neutron.
In conclusion, our theory predicts that, for fixed values of V0' and l, the neu
tron could be produced by the fusion of the proton and the electron.
This result is so important that it does not require any other comment, but, in
stead, it requires an urgent proposal of experimental confirmation.
From an experimental view point, we have the opinion that some different experi
mental arrangements could be conceived. We propose, here, some different indica
tions but our proposal should be intended as the occasion to begin a debate amo
ng the researchers in order to define a good strategy of experimentation.
As the first stage of the experimentation, one must provide the protons and the
electrons. The simplest way is the ionization of the hydrogen. In this case, t
he recombination of the proton with the electron to form the hydrogen atom must
be avoided, and a considerable number of protons must collide with the electro
ns and their contact will secure the fusion of these two particles. Note that t
he recombination of the proton with the electron to form hydrogen usually occur
s in time about 108 sec. Thus, we need a high frequency ionizing electromagnet
ic field in order to produce a kind of plasma. The use of a magnetron, as well
as of a laser, could be appropriate. In addition, let us observe that our exper
iment requires the proton to be at rest and the electron having 0.800 MeV at le
ast.
Let us examine, now, another experimental problem. We have seen that in the val
ue fixed by (5.5), almost 60 keV should be produced for each fusion. Thus, in p
rinciple, it should not be difficult to accertain that this energy is produced.
Some particular difficulties, instead, should regard the revelation of the prod
uced particles as result of the fusion. If a neutron is produced, it is quite c
lear to affirm that this particle could not be detected by usual detectors in t
he case of the presence of an high frequency ionizing field.
The presence of the e.m. field should disturb the electronics employed for the
detection. An alternative method could be represented from the use of the neutr
on activation. Some selected elements could be sealed in the proximity of the i
nduced fusion and the induced activation could be measured, soon after the fusi
on, by gamma spectrometry.
A second method of analysis and detection of the neutrons could be employed whi
ch uses appropriate dosimeters based on thermolumenescence, with the possibilit
y to select, unequivocally, the presence of the neutrons.
Finally, the last possibility could be given by using plastic or other passive
detectors usually employed in the dosimetry of the neutrons.
The strategy of the revelation of the produced neutron must be selected also co
nsidering that the observation of the created neutron is complicated by the pre
sence of a natural neutron background. This problem could be enhanced from a po
ssible low cross section for our neutron formation.
In this manner, we have given a preliminary indication of the problems that are
involved in an experimental verification of the proposed process of fusion. A
contribution to the debate is expected in order to optimize actually the experi
mental strategy.
THE SPIN OF THE CREATED NEUTRON
We have to calculate, now, the spin of the neutron created as bound st
ate of the proton and electron.
Consider that Rutherford (Rutherford, 1920) was the first to make the historica
l hypothesis on the neutron as a compressed hydrogen atom. This model was aband
oned since it resulted impossible by energetic balancing considerations and, in
particular, it resulted incompatible with the usual quantum mechanics which wa
s formulated in its original arrangement in 1927. In this paper we have seen th
at Rutherford's model of the neutron is not incompatible with our biquaternion
quantum mechanics; on the contrary this model is admitted from our theory in vi
ew point of energetic considerations as well as of the generalized quantum prin
ciples that our theory contains.
In the final part of this paper we will show that, according to Rutherford's hy
pothesis, the spin of the neutron in the model (p, e) coincides with the spin
of the proton (see also Conte, 1994c, 1995a).
To complete this argument we have to mention here that other authors (Santilli,
1990, 1993) obtained a direct theoretical evidence of Rutherford's historical
hypothesis on the neutron. This author also discussed in detail the problem of
the spin. Santilli's formulation (for details, see Santilli 1990) has some cont
act points with our formulation (in particular the (3.16) and the (5.2)), but t
he basic differences between Santilli's formulation and our theory have been di
scussed in detail in our previous papers (Conte, 1994c).
Now, let us explain our elaboration on the 1/2 h spin of the neutron conceive
d in accord with Rutherford's hypothesis. It was shown by us (Conte, 1994c) tha
t, according to the (2.8), new Pauli's matrices are possible in biquaternion qu
antum mechanics.
In particular, it was shown that for
X
d
RSymbol
`
Z
X
d
RSymbol
`
one obtains that
Ed
Symbol
`
rEd
Symbol
`
From Sz = 1/2 h e3, we have the new variable of spin
k
`
`
`
H
MrEd
Roman
`
`
`
According to the rule n. 7 of biquaternion quantum mechanics (Conte, 1994c),
n
rEd
rEd
9
`
9
`
where P1(l) and P2(l) are the probabilities for the two possible states of the
electron, and l1 and l are the two possible eigenvalues. Thus, we have that
`
rEd
TOOEM
`
rEd
We see that in biquaternion quantum mechanics the notion of spin leaves off bei
ng the rigid and immutable notion of the usual quantum mechanics. On the contra
ry, as it may be deduced from the (6.5), the biquaternion quantum mechanics gen
eralizes the notion of spin, showing that the usual values 1/2 h for this var
iable, are obtained from the general case for l = 1. Now, we have to clear the
reason that induces us to use the (6.5) for l # 1, instead of the usual case l
= 1, for our electron.
Let us remember that we are studing the penetration of the electron into the pr
oton conceived as a massive nonpoint like particle with a medium represented b
y the (5.2). The electron changes its physical characteristics penetrating the
proton: we have seen in the (5.4) that its energy is changed in accord to the b
asic features of the spacetime in which the electron penetrates; in the same m
anner, the electron changes the spin. Both (5.2) and (6.5) have been derived by
us for the first time on the basis of the (2.8). It generalizes the usual quan
tum mechanics and the usual Minkowski spacetime that was conceived only for th
e vacuum and, thus, does not represent the proton that, instead, we consider no
npoint like and with an extended medium.
We have to consider, now, the angular momentum of the electron.
Before of all, let us express the angular momentum in classical terms.
Consider the following two biquaternions
Z1 = x1e1 + x2e2 + x3e3 ; Z2 = p1e1 + p2e2 + p3e3
(6.6)
The angular momentum L is the biquaternion
L = Z1Z2  A = i e1L1 + i e2L2 + i e3L3
(6.7)
r = (x1, x2, x3) ; p = (p1, p2, p3) ; L = (L1, L2, L3) ; A = the scalar quantit
y (x1p1 + x2p2 + x3p3). r, p, L have bee refered here to the internal space (e1
, e2, e3) of the biquaternions.
The norm of L is
N (L) = L12 + L22 + L32
(6.8)
as it is required in our usual Euclidean metrics, with L2 = L12 + L22 + L33.
.
To describe the angular momentum of the electron penetrating into the proton, w
e have to transform L for the proper spacetime of the proton, and to fix the q
uantum properties of L.
To calculate (5.2), we have used (2.8) to change the description from vacuum to
the medium that has bm as characteristic functions. In consequence, for r we h
ave obtained the change in r' with xm' = bm xm, for p, we have obtained the ch
ange in p' with pm' = bm pm, for L we have the change in L' with Lm' = bm Lm
(Conte, 1995).
Note that the new spacetime with bm# 1 has a new metrics that is now completel
y different from the usual Euclidean metrics. For the generic vectorial physica
l quantity A (A1, A2, A3) we have A2= A1A1+ A2A2+ A3A3 in the case of the Eucli
dean space, and, instead, we have A2= A1b12A1+ A2b22A2+ A3b32A3 in our spaceti
me with characteristic functions bi, i = 1, 2, 3.
Also the properties of rotational symmetry are now changed in our spacetime fr
om being
inv. = x1x1+ x2x2+ x3x3
(6.9)
in the ordinary space, to become
inv. = x1 b12 x1+ x2 b22 x2+ x3 b32 x3
(6.10)
in our considered spacetime.
After this explanation on the physical features of the spacetime that we have
in consideration, we are now in the condition to conclude that, in the passage
from the classical case to the quantistic case, the eigenvalues of the operator
s L' 2 and L'3 are given by f (b1, b2, b3) l [ f (b1, b2, b3) l + 1 ] and f (b1
, b2, b3) m1, respectively, where l (l + 1) and m1 are the well known wigenval
ues of the angular momentum observables L2 and L3 of the usual quantum mechanic
s.
f (b1, b2, b3) is a function of bi (i = 1, 2, 3) to be calculated in the partic
ular case taken in consideration.
We are now in the condition to conclude this brief discussion on the spin of th
e neutron. Considering the (6.5) and the calculated values for the angular mome
ntum of the electron, we have that the total spin of the neutron may coincide w
ith the spin of the proton when the electron penetrates into the proton. In fac
t, we have that
Sn = Sp + f (b1, b2, b3) + 1/2 h l1
(6.11)
or
Sn = Sp + f (b1, b2, b3)  1/2 h l
(6.12)
For
l =  h / [ 2 f (b1, b2, b3) ]
(6.13)
or
l = 2 f (b1,b2, b3) / h
(6.14)
we have that
Sn = Sp
(6.15)
and Rutherford's hypothesis is fully confirmed.
In conclusion, as well as possible, we have solved all the problems that appear
to characterize the theoretical problem of the fusion of the proton and the el
ectron to form the neutron. Our aim is that an analogous effort will be possibl
e at the experimental level in order to confirm an experience that should be of
historical value.
REFERENCES
Conte E. (1993), On a Generalization of Quantum Mechanics by Biquaternions,
Hadronic Journal 16, 261
Conte E. (1994a), An example of Wave Packet Reduction using Biquaternions,
Physics Essays, 6, 4
Conte E. (1994b), Wave Function Collapse in Biquaternion Quantum Mecha
nics, Physics Essays, 7, 14
Conte E. (1994c), New Pauli's Matrices in Biquaternion Quantum Mechanics,
in publication on Physics Essays (the last num
ber, Decem
ber 1995)
Conte E. (1995), On a Generalization of Schrdinger's Equation using Biqua
ternions, Physics Essays, 8, 52
Conte E. (1995a), On a Generalization of Physical Laws by using Biquater
nions: the case of Minkowski's spacetime, in
publication
on Physics Essays;
A new Theory of Isospin in Biquaternion Quantu
m Mecha
nics, in publication on Physics Essays.
For a detailed exposition of all the arguments of the theory, see the book:
Conte E., Meccanica Quantistica Biquaternionica, Publishers Casa Editrice Pitag
ora, Bologna Italia.
D'Espagnat B. (1977), Conceptual Foundations of Quantum Mechanics, W. A.
Benjamin Inc.
Fermi E. (1947), Nuclear Physics, Chicago
Flugge S. (1971), Practical Quantum Mechanics, Springer Verlag, N. Y.
Janussis A., Mijatrovic M., Velianoski B. (1978), Physics Essays, 4, 2, 25
Mickens R.E. (1979), Long Range Interactions, Foundations of Physics, 9, 261
Philips J.C., Kleinman L. (1959), New Method for Calculating Wave Functions
in Crystals and Molecules, Ph
ys. Rev., 116, 287
Rutherford E. (1920), Proc. Roy. Soc. A97, 374; see Segre E.(1966), Nuclei
e Particelle, Zanichelli, Bologna, Italy
Santilli R. M.(1990 and 1994), Elements of Hadronic Mechanics, The Institute
.
for Basic Research, Palm HarborUSA
usual quantum mechanic
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