Application of correlated states of interacting particles in nonstationary and

modulated LENR systems

Vladimir I.Vysotskii Kiev National Shevchenko University, Kiev, Ukraine

Electrodynamics Laboratory "Proton-21", Kiev, Ukraine

Mykhailo VysotskyyKiev National Shevchenko University, Kiev, Ukraine

Stanislav V.AdamenkoElectrodynamics Laboratory "Proton-21", Kiev, Ukraine

In the work the most universal mechanism of optimization of low energy nuclear reactions on the basis of correlated states of interacting particles is considered. This mechanism provides giant increase of barrier penetrability under critical conditions (very low energy, high barrier), where the effectiveness of "ordinary" tunneling effects is negligibly small, and can be applied to different experiments. The physical reason of the barrier penetrability increasing in correlated states is connected with the modified uncertainty relation for correlated states.

In the work the preconditions and methods of formation of correlated coherent states of interacting nuclei are discussed.

It was shown for the first time that in real nuclear-physical systems very sharp grows (up to and more times!) of Coulomb barrier penetrability at very low energy (e.g. at E0.025 eV in partially correlated states of interacting particles is possible. Several successful low-energy correlated-induced fusion experiments are discussed.

30 50 200 100010 ...10 ...10 ...10

Uncorrelated states of particles and Heisenberg -Robertson uncertainty relation (1927)

The traditional approach to the physics of charged particles tunnelling is based on the assumption of mutual independence of the particle states corresponding to each energy level of quantized states. For such systems the tunnelling processes for each state are also mutual independent. Atomic and nuclear physics use widely the well-known Heisenberg uncertainty relation

which connects the dispersions and mean square errors of the coordinate q and the corresponding component of the momentum of a particle p.

2 2

2

/ 4, / 2; ( ) ,

( ) , ,

p q p

q p q

p q p p

q q p q

Generalized uncertainty relation

was discovered by Robertson in 1929 for mutual independent dynamical variables A

and B, whose commutator is nonzero.

This relation follows from analysis of base expression

at condition that the parameter is purely real.

0|)()(| 2 dqqivquG

);()()( qAAqAu

)()()( qBBqBv

2 2| [ ] | / 4, ( )A B AAB A A

2 2 * *( ) ( ) ( ) ( ) ( ) ( ) ,

( ) ( ) ( ), ( ) ( ) ( ).

A B u q u q dq v qv q dq

u q A A q v q B B q

2 arbitrary| ( ) ( )| 0 real parameter,G u q iv q dq

Heisenberg -Robertson uncertainty relation (1927)

Analysis of base expression:

2 22 0

2 4K K

G D K C D CD D

* * *

*

( ) ( ) , { ( ( ) ( ) ( ) ( )) },

( ) ( )

D u q u q dq K i u qv q u qv q d

rea

q

C v qv q l vadq lues

2

2* * * *

/ 4 :

1( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ))

4

C K D

u q u q dq v qv q dq u qv q u qv q dq

2 2 * * 21

( ( ) )( ( ) ) | ( )( ) ( ) |4

AB BAA B q q dq

2 2 21

( ( ) )( ( ) ) | [ ] |4A BA B AB

* *

* * *

( ( ) ( ) ( ) ( ))

( ( )) ( ) ( ) ( )

u qv q u qv q dq

q q dB q q qA dqA B

* * * *

* * *

( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( ) .

A q q dq qB B A

A B

q dq

B q q dq A q q dq

Generalized Heisenberg uncertainty relation

Relation can be used for the estimation of barrier transparency

L(E) – width of the barrier V(q)

- mean square

effective radial momentum of a particle

with energy EV(q) in the subbarrier region V(q) E, 0 q L(E).

For such system uncertainty relation has the form

In the case of low energy the condition is satisfied, then the transparency coefficient of the Coulomb barrier will be extremely small:.

/ 2p q

1}/)(2exp{}])([2)/2(exp{)()(

ELpdqEqVMEDELR

R

;)(EWeD

2 ( )p M V q E

( ) / 2L E p

/ 2 ( )p L E

}/)(2exp{])([2)/2()()(

ELpdqEqVMEWELR

R

E V

V(q)

W

R R+L(E) 0 q

E

In 1930, Schrödinger and Robertson independently generalized the Heisenberg idea of the quantum-mechanical uncertainty of different dynamical quantities A and B on the basis of the more correct analysis of base expression

, ,

.

If we remove the ungrounded limitation that the parameter is purely real, then yields the more universal condition called the Schrödinger--Robertson uncertainty relation.

- coefficient of cross correlation

cross dispersion of A and B

0|)()(| 2 dqqivquG

)()()( qAAqAu

)()()( qBBqBv

BAABr /

BAABBABA

AB 2

)(

2

},{

2

| [ ] |;

4(1 )A B

AB

r

1 1r

Correlated coherent states of particles and Schrödinger-Robertson uncertainty relation (1930)

From Schrödinger--Robertson uncertainty relation follows

2

2 2

2 2

2

;22 1

( )2

( ) ( )

0, 02

,2 1

eff

effeff

p qr

qp pqq p

rp p q q

qp pqr if q p

p q

E tr

| ,| 1 effp qr At we have

2 2 * *( ) ( ) ( ) ( ) ( ) ( ) ,

( ) ( ) ( ), ( ) ( ) ( ).

A B u q u q dq v qv q dq

u q A A q v q B B q

2 arbitrary com| ( ) ple( )| 0 x param r, eteG u q iv q dq

Shredinger -Robertson uncertainty relation (method of calculation)

2 * * *

2 * * * *

( ) ( )

( ) ( ) 0.

G u udq i u v uv dq

u udq u v uv dq v vdq

* * 2 * * 22

2 2

2* *2

2

2* *2

2

( )| |

2 | |

( )| |

{ ( ) } { ( ) }| | 0.

4 | |

2 |

|

|

4 |

G

i u v uv dq u v uv dqv

i u v uv dqu dq

u dq

u v uv dqu dq

u

dqu dq u dq

dq

2 2 * * 2 * * 21 1| | | | { ( ) } { ( ) }

4 4u dq v dq i u v uv dq u v uv dq

At

we need

* * 2( ) / 2 | |i u v uv dq u dq

* * 2( ) / 2 | |u v uv dq u dq

,

)

2 22 | [ ] |

( ) 14

ABA B AB A B

A B

AB

( )

2AB

ABA B

AB BAA B r coefficient of

correlation

2

2| [ ] |

, 1 14(1 )

A B ABAB

ABr

r

For Coulomb potential barrier the modified uncertainty relation is

At full correlation |r|1 the mean square effective coordinate of a particle will be unlimited ( ) at any energy!

In this ideal case the tunnel transparency of arbitrary potential barrier will be close to 1 at any low energy E of the particle (!):

1||1)}(exp{ /)(1 2

rateEWD qELr

2

2

22 1

2 1 8 ( )

effqpr p

r M V q E

V(q)

W

R R+L(E) 0 q

E

q

The physical reason for the increase of the probability of tunneling effect is related to the fact that the formation of a coherent correlated state leads to the cophasing and coherent summation of all fluctuations of the momentum

for various eigenstates forming the superpositional correlated state. This leads to a very great dispersion of the momentum

and very great fluctuations of kinetic energy

of the particle in the potential well and increasing of potential barrier penetrability.

( ) ( )N

nn

p t p t

2

2 2 2{ ( ) ( ) } ( ) ( )N

p n n n mn

p t p t p t N p N p p

2 2 2 2( ) / 2 / 2 ( ) / 2 ~n m nT p t M N p p M N p M N

1np

1np

np

Uncorrelated state

2 2( ) ( )uncorr n nn

p p N p

1np

1np

np

2 2 2( ) ( ) ( )corr n n m nn

p p N p p N p

Correlated state

Another physical reason for an increase in the transparency in the superpositional correlated state consists in the following. The presence of a partial correlation of different eigenstates forming the super-positional correlated state causes the partial mutual damping (compensation) of all partial waves reflected from the walls of the potential well (back destructive interference), which leads to a interferometric increasing of potential barrier penetrability. At the full correlation of all eigenstates, there occurs the full damping of reflected waves, which gives the complete transparency of the barrier at any low energy.

Such interference is absolutely impossible for uncorrelated states.

Formation of a correlated coherent state of a particle

One of the simplest methods of the formation of the correlated coherent states of a particle is connected with parametric pumping of nonstationary harmonic oscillator with non-stationar Hamiltonian

being firstly in the ground state, to coherent correlated superposition state.In this case the solution of Schrödinger equation is the system of normed eigenfunctions

}2/]||8)/(]/[exp{[1

),(),( 2*20

2

4 2

dtdittq

Here, = q/qo is the normalyzed coordinate, is the initial frequency of the harmonic oscillator, is any constant complex-valued number;(t) is a complex solution of the classical nonstationary equation of motion of the oscillator with variable frequency (t).

This solution satisfies the condition

,/ 00 Mq )0(0 t

0)(/ 222 tdtd

idtddtd 2// **

2 22 2

2

( )( , )

2 2rM t xd

H x tM dx

and compression coefficient of the correlated state

2

202

*

|/|1,

|/|

/Re

dtdr

dtd

dtdr

The use of the same function leads to the following formulas for the correlation coefficient

),( tq

|/|

|| 0

dtdk

p

q

2 2/ 2 (1 ), / 2 (1 )q pk r k r

At |r|1 we have

For Cartesian coordinates

2 2,

2 1 2 1q p

r r

0

20

2

22 11

1,x

xM r

rr

00

2 2

2

2 1, 1

1x

x

pp r

M

r r

2

0 24

1 ( )( , ) exp 1

42 1 ( )xx

x ir tx r

r t

The wave function of the coherent correlated state of any particle in nonstationary

harmonic oscillator with concrete correlation coefficient is

t

rr

dr

tr

trtg

dt

dr

r

tg

rtgAt

0202

0

22)(1

)(2

|)(|

)()(,

1

)(1

)1())((

1)(

BAAB ttr /)()(

The connection between the given change of the correlation coefficient r(t) and the

necessary law of variations in the actual frequency of the harmonic

oscillator:

)(tr

0

0 00 02

0

2 | |1. ( ) / 1

1r

r tAt t we have r const

r

0( )( ) | ( ) |

0, | ( ) | 02. rrAt we had

vtd t d r t

r tdt dt dt

e

0( )( ) | ( ) |

0, | ( ) | 13. rrAt we had

vtd t d r t

r tdt dt dt

e

0

1.0

r(t)/(0)

0 t

r0r0(t)

0

1.0

|r(t)|

0 t

r0

3

1

2

Let's consider simple model which allows to demonstrate the process of barrier transparency increase during the increase of correlation coefficient. Let us suppose that the particle are presented in the deforming parabolic well, and the characteristic frequency of that well decreased to the value

to the moment t=0 in a specific way, and at that moment the particle state was characterised by the correlation coefficient After that moment (at t> 0) well deformation stops.For such moment

Change of the probability of the barrier transparency for particles in coherent correlated state.

( )r t0 ( 0)r t

sr

2

24

1exp 1

42 1( , , 0)corr s

s

xx s

x it

r

rx r

From the other hand the standard wave function of the particle in stationary well is

2 20/ / 2( , , 0 ( ( ( )) ) ) , ,niE t x x

n s n n n nn

s B r x e x ex Cr Ht x

20 0 0 0 0

0

1, / , ( 1/ 2), 2

2 !n n x

nC x M E n x

x n

Population coefficients

can be find from the condition of continuity for the total wave function at t0 and t0

*( , , 0) ( , , 0) ( ) ( )exp( / ),s corr s n n nn

x r t x r t x dx x iE t

The resulting expression for the full wave particle function in partially correlated

state with rs in the stationary well at t0 has the form

*( ) ( , , 0) ( )n s corr s nB r x r t x dx

The results of calculations of probability density of particle localization within a

parabolic well at any moment of time t0 depending on the correlation coefficient r0

are presented below.

0t

x/x0

x0

0t

x/x0

x0

Changes of the cross structure of the distribution in an uncorrelated state (a) and in a state with the coefficient of correlation (b) in the course of time.

2| ( , , 0) |sx r t 0sr 0 0.9r

0.15

0.10

0.05

c)

0t0 1 2 3 4

5 6

x0

r0 =0.98

r0 =0.95

Rs =0.9

rs=0.5

rs=0

Periodic modulation of probability density for states with the coefficient of correlation r0=0;0.5;0.9;0.95;0.98 in the same potential well for different values of transversal coordinates x in V(x) at: a) x=0; c) |x|=2x0

20| ( , , 0) |x r t

1.5

1

0.5

r0=0

r0=0.5

r0 =0.9

r0=0.98

r0 =0.95

x0a)

0t

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

50-5

rs=0.98

rs=0.95

rs=0.9

rs=0.5

rs=0

x/x0

delocx0

V(x)

x/x050-5-10

1

2

3

rs=0

rs=0.5

rs=0.9

rs=0.95

rs=0.98

locx0

V(x)

Change of the cross structure of the distributions and in states with the coefficient of correlation r=0.5;0.9;

0.95; 0.98 and in an uncorrelated state with r=0 at the time moment corresponding to the maximal and minimum spatial variance of the coordinate in an area under the barrier

2| ( , , ) |s deloc delocx r t 2| ( , , ) |s loc locx r t

20( ) ( / )V x x x

Averaged distributions of the probability density for a particle in the same potential well in an area under the barrier for correlated and uncorrelated states of particles with r=0;0.5;0.9;0.95;0.98 .

2| ( , , 0) |s tx r t rs=0.5

rs=0.9

rs=0.98

-6 -4 -2 0 2 4 6

x/x0

<>x0

0.5

0.4

0.3

0.2

0.1

rs=0V(x)

rs=0.95

Change of the barrier transparency

for selected coordinates under the same parabolic barrier V(x) for different coefficients of correlation.

2| ( , , 0) |s tx r t ( )nx

The final change of the probability density (barrier transparency)

2 20

1| ( , , 0) | | ( , , 0) | , 2 /

t T

s t s

t

x r t x r t dt TT

The very sharp grows (up to and more times) of the barrier transparency

and the same grows of rate of nuclear synthesis (averaged probability of reaction per unit of time for one pair of interacting nuclei situated at )

for different distances under the barrier with increasing of the coefficient of correlation is related to the Schrödinger—Robertson modified relation of uncertainty and is the direct result of this relation.

30 4010 10

( ) ( ) 20( ) | ( , , ) |n n

tx x r t

( ) 20 2

( )| ( , , ) | ,

s

nr t

S Ex r t

Me

( )nx

12 12

25 39 30

5 20 30

16 30

16 30

12 12

3.5*10 * 10 / ;

( 2.6*10 * 5*10 / ;

0.11 * 1.5*10 / ;

8.8*10 * , 20

( , )

)

(

) /

)

(

pp

pd

pd

C C

For S MeV bn and cm s

S MeV bn and cm s

p p reaction

p d reaction

d d reaction

C C r

S MeV bn and cm s

S MeV bn ceaction m s

12 16

16 16

21 5 30

28 12 30

12 16

16 16

;

1.2*10 * , 2.3*10 / ;

2.3*10 * , 3.9*

)

( ) 10 /

(C O

O O

S MeV bn cm s

S MeV bn cm

C O reaction

O O reaction s

2 3 2 / 3( ) [1/ ( / ) ] , 2 ( 1) , 10( 2)F t c c t c case c case

0 2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1.0

1 2

b)

r(t)/0

t/

Change of correlation coefficient r(t) of the particle in nonstationary harmonic oscillator (a) connected with the change of frequency of the oscillator (b) for function

0 2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1.0

1 2

a)

t/

r(t)

( )r t

2

2 3 2/3( ) [1/ (/

( ) , /) ( / )

) ](

F tt

r t c tt

cF t

Formation of correlated states at monotonous modulation

30 40 5010 20 0t

1.0

0.5

0.6

0.8

0.9

0.7

0.4

0.3

0.1

0.2

0.0

|r|

0/=/2

30 40 500 10 20 0t

1.0

0.5

0.6

0.8

0.9

0.7

0.4

0.3

0.1

0.2

0.0

|r|

0/=

1.0

0.5

0.6

0.8

0.9

0.7

0.4

0.3

0.1

0.2

0.0

30 40 500 10 20 0t

|r|

0/=(3/2)

Time dependences of the correlation coefficient r for a periodic variation in the oscillator frequency

various ratios of and

Excitation of the correlated states of a particle at harmonic modulation

00( ) cost t

Excitation of the correlated states of a particle at limited modulation

0( ) (1 cos )t g t

0.5

0.6

0.8

0.9

0.7

0.4

0.3

0.1

0.2

0.0

1.0

60 80 0t0 20 40

c)|r|

Time dependences of the correlation coefficient r for a limited change in the oscillator frequency

at 0 0.3and g

rD

21 ( )1( ) ,

2

tr t

r r o

t

D D dt

0 20 40 60 80 0t

8010

6610

5210

3810

2410

1010

1,r rD D

a)

rD 0( ) (1 cos )t g t

0 / 2 0.1and g

Time dependences of real and averaged coefficient of transparency for the case of oscillator frequency

at

1000 2000 3000 4000 ot

-1.0

-0.5

0.5

1.0

3985 3990 3995

0.5

1.0

3997.53998.0 3998.5 3999.0

0.99992

0.99994

0.99996

0.99998

1.00000

-0.5

-1.0

0

r(t)

ot

ot

r(t)

r(t) 0

0

( ) (1 cos ),

0.1, /100 ,

t g t

g

max

4

!1 2

2440

max

4| | 0.999

1059

998 | |

0

1000 , / 5.10 ,

: 28, 1

0 : 5.10

| | 0.999998 :

0.01, 10

A

r

r

A

r

r D e

T K v c

Ni p Cu Z Z

r

D D

0.999998

Conclusions

Presented results clearly demonstrate the "giant" increases (by many order of magnitude) of localization density under the potential barrier and also the possibility of very effective under the barrier penetrations of particles at the increase of correlation coefficient (up to and more times).

Such effects take place in different nonstationary LENR experiments with release of energy (e,.g. action of ultrasound, the method of “SuperWave”, experiments of Ross and experiments on isotope transmutation in growing (non-stationary) biological systems).

1000100 10...10

One more nontrivial result seems to be important. At increase of the correlation coefficient r1 the giant increase the dispersion of the nucleus coordinate (q) takes place. At such condition the mean square uncertainty of the of this nucleus coordinate can significantly exceed the mean distance between nuclei <a>.In this case, the volume of the three-dimensional region, where a nucleus in the correlated state is localized, will contain

nearest nuclei, with which a nuclear reaction is possible.

This result opens the way to the possibility for correlated collective many-nucleus reactions to run. The probability of such reactions increases by Nc times as compared with the probability of the running of binary reactions in the absence of a correlation.

qq

3( / ) 1c qN a