- Email: [email protected]
On the Barut-Vigier model of the hydrogen atom
9 September1996 PHYSICS
LETTERS
A
Physics Letters A 220 (1996) 297-301
IELSEVIER
On the Barut-Vigier
model of the hydrogen atom
N.V. Samsonenko, D.V. Tahti, F. Ndahayo Department
of Theoretical
Physics,
Russian
Peoples’
Friendship Uniurrsity, Russian Federation
Miklukho-Maklaya
Street 6, Moscow
117198,
Received22 May 1996;acceptedfor publication27 June 1996 Communicatedby P.R. Holland
Abstract 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.
1. Introduction The essential idea proposed by Vigier to explain the confirmation with H,O of a set of (so-called) “nuclear cold fusion” experiments [l] rests on the fact that the quantum electron jumps in the hydrogen atom (or deuterium, tritium, etc.) from Bohr’s usual orbits with energies = 14 eV into new “tight” quantum orbits with energies = few keV (as discussed in Ref. [2]). One can try to demonstrate the existence of the new levels by introducing explicitly magnetic spinspin and spin-orbit interactions. Through a study of exact solutions of the Dirac equation [3] Barut obtained a Schriidinger equation fiq= Eyr with a potential V(r) = A/r + B/r* + C/r3 + D/r4. This result was used by Vigier to analyze the problem of the hydrogen atom with a potential different from the Coulomb one. In the present work we shall make an attempt to analyze the results obtained in Ref. [2]. It is shown that when one neglects the tensor part of
bhe forces and one limits oneself to the central spin-spin and spin-orbit forces one can write explicitly the analytical form of the radial factor of the wave function. One can then try to improve the model through a more exact study of the influence of the non central part of the interaction Hamiltonian.
2. The Barut-Vigier assumption of new “tight” energy levels
As a starting point of the Baru-Vigier model one assumes that the system of two charged spinning particles in the non-relativistic approximation can be described by the Pauli equation with a Hamiltonian which contains explicitly spin-spin and spin-orbit interactions, i.e. Hz-
I 2m,
rl -r2
P,+M,x C
0375.9601/96/$12.00 Copyright 0 1996Publishedby Elsevier ScienceB.V. All rights reserved. PII SO375-9601(96)00532-4
of the existence
IT,-r213
2
1
f-
Pz - e,M,
pm2
x
Taking into account the properties of a mixed product of three vectors we can then write H in the form
f-z-f-1
I rl - r2 I 3
C
2
e1e2 -Ml +
I rl
-
r2
-M2421r,
-rz).
I
(1)
Here m,, P,, e,, r,, M, are the mass, the momentum, the charge, the radius and the magnetic dipole of the first particle; mz, P2, e2, r2, M, are the same properties of the second particle. From the very beginning of our analysis, as in Ref. [2], we shall neglect the last term in (1). Let us Anow look for the solution of the Pauli equation H?lJ = EV, where 6 is given by relation (1). To do that let us introduce the following notations, P=P, r=r,
+p2,
M,=m,
-km,,
17213112
-r2,
lu=
2~ i-
r
ete,2h2
2c2m,m2
r’
1
This formula differs from the one given in Ref. [2] by the two last terms. This difference is due to the neglect in Ref. [2] of the magnetic moment of one of the two particles. Since the operators i? and ( 6. t) commute, i.e. they have common eigenfunctions, one can replace the operator ( 6. fl) by its eigenvalue CT. Then one obtains
m, -k-m,’
m2P, -6
&&‘--
MO
.
We then pass to coordinates, tied to the center of mass, by writing P = 0. Hamiltonian (1) then takes the form 2 e,M2
H=E-p 2P
(f7.L)
4c’pm, m2 14 ’
-2
P=
e,e,h
e1e2
H=P+--
-+cm1
+-.ele2
e2M,
cm2
xi
r3
ele2
e,e,fia
r
2c%n,ln~,3
eZe21i2 12 +
4c4pm, pz r4 . (4)
There is a set of contradictory estimates of energy values and quantum orbit-radii based on Hamiltonian (4). Hence the analytical and exact solution of the Pauli equation with Hamiltonian (4) is of great use. A similar Hamiltonian can be obtained [4] for a nonrelativistic charge e, moving in the vector potential A = p2 CTX r/r3 of another charge e2 with magnetic moment p, and p2 (see the Appendix).
(2)
r
One can assume for simplicity that each particle possesses a normal magnetic moment M, = (e,h/2m,c)a, and M, = (e,h/2m,c)a,. In this case the terms of the Hamiltonian can be written in the form
3. The analytical solution of the radial Schriidinger equation with the Barut-Vigier potential
Our purpose now is to solve the stationary equation $9 = E9, where e2e2ii2 12
e,e,fia
2c2m,m2r3
+
4c4pm,m2r4. (4)
e2e2h2 I 2
1
= 4c4pm,m2 -7. r
Solving that equation in spherical coordinates one finds that the wave function has the usual angular
N-V.
Samsonenko
cependence and the radial function equation 2pr2
,(&‘)‘+-$-
E-y!e2e2h2 I 2
R=O.
- 4c4pm,m2r4
I&t us assume that X = rR. Then, remarking that (l/r2>(r2R’Y = (l/r)(rRy’ one finds that the function X satisfies the equation fi*qz+
Letrers
A 220
299
(19961297-301
One can rewrite X” -= X
2pr2
e,e,ho-
2c2m,m2r3
satisfies the
ii2Z( 1 + 1)
i +
R(r)
et ai./Physics
Eq. (5) in the form
4 --E+-+-f-+--, r
b, r2
ba r4
here E= (2p./fi2)E, b, = (2p/h2)e,e,, b, = 1(Z + b, = peeee,cr/c2hm,m2, b, = I>, efez/2 c4ml m2. To study the case n = 0, one uses (6) in (7) and obtains the following relation between constants b, = e2,
b,=28(a-l),
b,=a(a-I)-2K8,
1)
b, r3
b,=
-2Ck’K,
E=
(8)
2pr2 +
e2e2fi2 1 2
e,e,hm 2c2m,m2r3
- 4c4pm,m2r4
x=0.
(5)
1
‘?rom this equation it is evident that for long dis:ances (I -+ z) an asymptotic behavior X, of the i‘unction X can be found as a solution of the follow!ng equation,
!:ts solution is given by the formula -2pE
.&=AeCK’,
K=
h2
2c4m,m2r4
0
=
A-,-e-
e/r
46+2,i& 4
d
+b,+2&
- b, = 0,
(93
---&-(b~~~~)f
x’yr)
to the wave-function
= jppe-K’e-e/‘,
where N(O) is a normalization In the case n = 1 one has
x0 = 0.
o=
2blb4
which determines the parameter CT. However, one has to remember that as eigenvalue of the operator (a * L), (P can take integer or half-integer values. From (8) and (9) one obtains the energy eigenvalue
f(r)
)
4
That energy corresponds can be
One obtains X
b:
of the po-
’
,with E < 0. In the same way, the r + 0 asymptotic found as a solution of the equation
x;i -
and the relation between the coefficients tential
eo’=
-
d
-K2,
2 ele2
2c4m,m,
= (r-a\‘)),
and the relations between
2
constant.
the coefficients
take the
fOI-I?l
.
Applying the method proposed in Ref. [5] we will search for the solution in the form X(r)
=f(r)raXm(r)XO(r)r
where f(r)
= lfi(r-4n)).
n>O,
= 1,
n = 0.
+ b;;$-(&+a\1))-b2=0. (6)
(10)
300
N.V. Samsnnenko
et d/Physics
The energy eigenvalues are i _ _ 61 -
b? f i
b;/2fi)
4( &+
up>
*
Following Ref. [4] for a nonrelativistic charge e, moving in the vector potential A = p2 CTx r/r3 of another charge e2 with magnetic moment ,+ the Pauli equation
1. (11)
The corresponding
A 220 119961297-301
Appendix. Pauli equation for two charged particles with magnetic moments pI and p2
b; -4b,(/fojl))(l+ J
Letters
wave function is
In general one can use this algorithm to obtain the wave-functions for n = 2, 3,. . . . But even in the case n = 1 the method meets with difficulties because one can solve Eqs. (10) and (1 I> only numerically. In the cases n > 2 similar expressions are more complicated. However, the numerical solution is possible. For example by means of the MathCad computer program a “tight” state n = 1, 1= 0, E = 40 keV has been found in addition to the usual Coulomb states ‘. The detailed analysis of this model of hydrogen atom will be continued. Namely, we shall consider the interaction with an external electromagnetic field. A paper in which the main aspects of the method applied to solve the radial Schrbdinger equation for inverse-power potentials (i.e. main results obtained in [4]) are reexposed and comments on the numerical solution are made is to be published soon. Of course in this article a non-relativistic approximate calculation with the neglect of anomalous magnetic moments of particles was presented. But following a suggestion of Vigier a Pauliitype equation can be transformed into a relativistic FeynmanGell-Mann equation by the substitution of E+ T (proper time), xi --f x~, i.e. V2 into q .
for stationary states, with eii i-A.V=mc reduces to
752 y-/r+
The authors are thankful for valuable discussions with Professor J.P. Vigier, Dr. V. Mayorov and Dr. V. Muromtsev.
’ Such a state has already Szpak et al. [6].
been observed
using
deuterium
by
1
-c-2 2mr2
SS.L
where
Introducing J = L + S and expanding in spherical harmonics to separatethe angular variable, one has the radial equation fi2 -2,,Dr+
1(1+ l)Q 2mr2
-S[J(Jtl)-I(Z+l)-f]h’ +-
Acknowledgement
e1P2 -U-L, mcr2
4 14 + T-E 2mc2r4
U,,(r)
=O.
For a given sign of p,, say p2 > 0, the l/r3 term is negative only for I = J + + if e, < 0, or I = J - $ if el > 0. One expects therefore resonances for (J =- ;, Z=l),(J=+, r=2) ,... for e, 0. According to Vigier the new “tight” orbits are only possible in special types of dense media where they are built in special conditions under the influence of external magnetic fields, i.e. in electrolytic or discharge experiments.
N.V.
Samsonenko
et d/Physics
Letters
References [.I
M. Fleischman, S. Pons and M. Hawkins, J. Electroanal. Chem. 263 (I 989) 301. [7_] J.P. Vigier, New hydrogen (deuterium) Bohr orbits in quantum
[3] [4] [S] [6]
A 220
11996)
297-301
301
chemistry and “cold fusion” processes, paper, presented at ICCF 4, Hawai (3-6 December 1993). A.O. Barut and J. Kraus, J. Math. Phys. 17 (1976) 506. A-0. Barut, SUIT. High Energy Phys. 1 (1980) 113. S. kcelik and M. Simsek, Phys. Lett. A 152 (1991) 145. S. Szpak, P.A. Mosier-Boss and J.J. Smith, Phys. Lett. A 210 (1996) 382.
LETTERS
A
Physics Letters A 220 (1996) 297-301
IELSEVIER
On the Barut-Vigier
model of the hydrogen atom
N.V. Samsonenko, D.V. Tahti, F. Ndahayo Department
of Theoretical
Physics,
Russian
Peoples’
Friendship Uniurrsity, Russian Federation
Miklukho-Maklaya
Street 6, Moscow
117198,
Received22 May 1996;acceptedfor publication27 June 1996 Communicatedby P.R. Holland
Abstract 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.
1. Introduction The essential idea proposed by Vigier to explain the confirmation with H,O of a set of (so-called) “nuclear cold fusion” experiments [l] rests on the fact that the quantum electron jumps in the hydrogen atom (or deuterium, tritium, etc.) from Bohr’s usual orbits with energies = 14 eV into new “tight” quantum orbits with energies = few keV (as discussed in Ref. [2]). One can try to demonstrate the existence of the new levels by introducing explicitly magnetic spinspin and spin-orbit interactions. Through a study of exact solutions of the Dirac equation [3] Barut obtained a Schriidinger equation fiq= Eyr with a potential V(r) = A/r + B/r* + C/r3 + D/r4. This result was used by Vigier to analyze the problem of the hydrogen atom with a potential different from the Coulomb one. In the present work we shall make an attempt to analyze the results obtained in Ref. [2]. It is shown that when one neglects the tensor part of
bhe forces and one limits oneself to the central spin-spin and spin-orbit forces one can write explicitly the analytical form of the radial factor of the wave function. One can then try to improve the model through a more exact study of the influence of the non central part of the interaction Hamiltonian.
2. The Barut-Vigier assumption of new “tight” energy levels
As a starting point of the Baru-Vigier model one assumes that the system of two charged spinning particles in the non-relativistic approximation can be described by the Pauli equation with a Hamiltonian which contains explicitly spin-spin and spin-orbit interactions, i.e. Hz-
I 2m,
rl -r2
P,+M,x C
0375.9601/96/$12.00 Copyright 0 1996Publishedby Elsevier ScienceB.V. All rights reserved. PII SO375-9601(96)00532-4
of the existence
IT,-r213
2
1
f-
Pz - e,M,
pm2
x
Taking into account the properties of a mixed product of three vectors we can then write H in the form
f-z-f-1
I rl - r2 I 3
C
2
e1e2 -Ml +
I rl
-
r2
-M2421r,
-rz).
I
(1)
Here m,, P,, e,, r,, M, are the mass, the momentum, the charge, the radius and the magnetic dipole of the first particle; mz, P2, e2, r2, M, are the same properties of the second particle. From the very beginning of our analysis, as in Ref. [2], we shall neglect the last term in (1). Let us Anow look for the solution of the Pauli equation H?lJ = EV, where 6 is given by relation (1). To do that let us introduce the following notations, P=P, r=r,
+p2,
M,=m,
-km,,
17213112
-r2,
lu=
2~ i-
r
ete,2h2
2c2m,m2
r’
1
This formula differs from the one given in Ref. [2] by the two last terms. This difference is due to the neglect in Ref. [2] of the magnetic moment of one of the two particles. Since the operators i? and ( 6. t) commute, i.e. they have common eigenfunctions, one can replace the operator ( 6. fl) by its eigenvalue CT. Then one obtains
m, -k-m,’
m2P, -6
&&‘--
MO
.
We then pass to coordinates, tied to the center of mass, by writing P = 0. Hamiltonian (1) then takes the form 2 e,M2
H=E-p 2P
(f7.L)
4c’pm, m2 14 ’
-2
P=
e,e,h
e1e2
H=P+--
-+cm1
+-.ele2
e2M,
cm2
xi
r3
ele2
e,e,fia
r
2c%n,ln~,3
eZe21i2 12 +
4c4pm, pz r4 . (4)
There is a set of contradictory estimates of energy values and quantum orbit-radii based on Hamiltonian (4). Hence the analytical and exact solution of the Pauli equation with Hamiltonian (4) is of great use. A similar Hamiltonian can be obtained [4] for a nonrelativistic charge e, moving in the vector potential A = p2 CTX r/r3 of another charge e2 with magnetic moment p, and p2 (see the Appendix).
(2)
r
One can assume for simplicity that each particle possesses a normal magnetic moment M, = (e,h/2m,c)a, and M, = (e,h/2m,c)a,. In this case the terms of the Hamiltonian can be written in the form
3. The analytical solution of the radial Schriidinger equation with the Barut-Vigier potential
Our purpose now is to solve the stationary equation $9 = E9, where e2e2ii2 12
e,e,fia
2c2m,m2r3
+
4c4pm,m2r4. (4)
e2e2h2 I 2
1
= 4c4pm,m2 -7. r
Solving that equation in spherical coordinates one finds that the wave function has the usual angular
N-V.
Samsonenko
cependence and the radial function equation 2pr2
,(&‘)‘+-$-
E-y!e2e2h2 I 2
R=O.
- 4c4pm,m2r4
I&t us assume that X = rR. Then, remarking that (l/r2>(r2R’Y = (l/r)(rRy’ one finds that the function X satisfies the equation fi*qz+
Letrers
A 220
299
(19961297-301
One can rewrite X” -= X
2pr2
e,e,ho-
2c2m,m2r3
satisfies the
ii2Z( 1 + 1)
i +
R(r)
et ai./Physics
Eq. (5) in the form
4 --E+-+-f-+--, r
b, r2
ba r4
here E= (2p./fi2)E, b, = (2p/h2)e,e,, b, = 1(Z + b, = peeee,cr/c2hm,m2, b, = I>, efez/2 c4ml m2. To study the case n = 0, one uses (6) in (7) and obtains the following relation between constants b, = e2,
b,=28(a-l),
b,=a(a-I)-2K8,
1)
b, r3
b,=
-2Ck’K,
E=
(8)
2pr2 +
e2e2fi2 1 2
e,e,hm 2c2m,m2r3
- 4c4pm,m2r4
x=0.
(5)
1
‘?rom this equation it is evident that for long dis:ances (I -+ z) an asymptotic behavior X, of the i‘unction X can be found as a solution of the follow!ng equation,
!:ts solution is given by the formula -2pE
.&=AeCK’,
K=
h2
2c4m,m2r4
0
=
A-,-e-
e/r
46+2,i& 4
d
+b,+2&
- b, = 0,
(93
---&-(b~~~~)f
x’yr)
to the wave-function
= jppe-K’e-e/‘,
where N(O) is a normalization In the case n = 1 one has
x0 = 0.
o=
2blb4
which determines the parameter CT. However, one has to remember that as eigenvalue of the operator (a * L), (P can take integer or half-integer values. From (8) and (9) one obtains the energy eigenvalue
f(r)
)
4
That energy corresponds can be
One obtains X
b:
of the po-
’
,with E < 0. In the same way, the r + 0 asymptotic found as a solution of the equation
x;i -
and the relation between the coefficients tential
eo’=
-
d
-K2,
2 ele2
2c4m,m,
= (r-a\‘)),
and the relations between
2
constant.
the coefficients
take the
fOI-I?l
.
Applying the method proposed in Ref. [5] we will search for the solution in the form X(r)
=f(r)raXm(r)XO(r)r
where f(r)
= lfi(r-4n)).
n>O,
= 1,
n = 0.
+ b;;$-(&+a\1))-b2=0. (6)
(10)
300
N.V. Samsnnenko
et d/Physics
The energy eigenvalues are i _ _ 61 -
b? f i
b;/2fi)
4( &+
up>
*
Following Ref. [4] for a nonrelativistic charge e, moving in the vector potential A = p2 CTx r/r3 of another charge e2 with magnetic moment ,+ the Pauli equation
1. (11)
The corresponding
A 220 119961297-301
Appendix. Pauli equation for two charged particles with magnetic moments pI and p2
b; -4b,(/fojl))(l+ J
Letters
wave function is
In general one can use this algorithm to obtain the wave-functions for n = 2, 3,. . . . But even in the case n = 1 the method meets with difficulties because one can solve Eqs. (10) and (1 I> only numerically. In the cases n > 2 similar expressions are more complicated. However, the numerical solution is possible. For example by means of the MathCad computer program a “tight” state n = 1, 1= 0, E = 40 keV has been found in addition to the usual Coulomb states ‘. The detailed analysis of this model of hydrogen atom will be continued. Namely, we shall consider the interaction with an external electromagnetic field. A paper in which the main aspects of the method applied to solve the radial Schrbdinger equation for inverse-power potentials (i.e. main results obtained in [4]) are reexposed and comments on the numerical solution are made is to be published soon. Of course in this article a non-relativistic approximate calculation with the neglect of anomalous magnetic moments of particles was presented. But following a suggestion of Vigier a Pauliitype equation can be transformed into a relativistic FeynmanGell-Mann equation by the substitution of E+ T (proper time), xi --f x~, i.e. V2 into q .
for stationary states, with eii i-A.V=mc reduces to
752 y-/r+
The authors are thankful for valuable discussions with Professor J.P. Vigier, Dr. V. Mayorov and Dr. V. Muromtsev.
’ Such a state has already Szpak et al. [6].
been observed
using
deuterium
by
1
-c-2 2mr2
SS.L
where
Introducing J = L + S and expanding in spherical harmonics to separatethe angular variable, one has the radial equation fi2 -2,,Dr+
1(1+ l)Q 2mr2
-S[J(Jtl)-I(Z+l)-f]h’ +-
Acknowledgement
e1P2 -U-L, mcr2
4 14 + T-E 2mc2r4
U,,(r)
=O.
For a given sign of p,, say p2 > 0, the l/r3 term is negative only for I = J + + if e, < 0, or I = J - $ if el > 0. One expects therefore resonances for (J =- ;, Z=l),(J=+, r=2) ,... for e,
N.V.
Samsonenko
et d/Physics
Letters
References [.I
M. Fleischman, S. Pons and M. Hawkins, J. Electroanal. Chem. 263 (I 989) 301. [7_] J.P. Vigier, New hydrogen (deuterium) Bohr orbits in quantum
[3] [4] [S] [6]
A 220
11996)
297-301
301
chemistry and “cold fusion” processes, paper, presented at ICCF 4, Hawai (3-6 December 1993). A.O. Barut and J. Kraus, J. Math. Phys. 17 (1976) 506. A-0. Barut, SUIT. High Energy Phys. 1 (1980) 113. S. kcelik and M. Simsek, Phys. Lett. A 152 (1991) 145. S. Szpak, P.A. Mosier-Boss and J.J. Smith, Phys. Lett. A 210 (1996) 382.