228(12) : LENR, Relativistic Theory of Quantum Tunnelling

Feed: Dr. Myron Evans
Posted on: Tuesday, September 25, 2012 3:23 AM
Author: metric345
Subject: 228(12) : LENR, Relativistic Theory of Quantum Tunnelling

In this case the transmission coefficient is given by eq. (10) and for a given particle mass m, barrier thickness 2a and potential height V sub 0, T can be plotted against the incoming particle velocity v. The new linear equation of relativistic quantum mechanics was used to derive this result. In the hyper relativistic limit:

v goes to c

the following assumption is usually used

gamma m = 1

In the non relativistic limit:

v << c

these equations reduce to the theory already graphed for UFT228. So Horst may like to graph these results as usual and I will proceed to write up UFT228 with co authors Horst Eckardt and Douglas Lindstrom.

a228thpapernotes12.pdf

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Low Energy Nuclear Fusion Reactors (LENR)

Feed: Dr. Myron Evans
Posted on: Tuesday, September 25, 2012 1:40 AM
Author: metric345
Subject: Low Energy Nuclear Fusion Reactors (LENR)

Many thanks to Dr Gareth John Evans. I hope that LENR will be developed with the utmost urgency to replace wind turbines. LENR should be used alongside gas and coal until new forms of energy production are ready to come on market.

Excellent new physics that the standard model could never explain. This may be how science progresses but increasingly we must wonder how much has been missed by the old physics with all its flaws. Modern physics was naive and full of errors and misconceptions that resulted at times in nonsense and deflected thought from areas like LENR that are important and useful. Well done both of you – your contributions to natural philosophy are immense.

Subject: Fwd: Transmission plots, E dependence

Co author of UFT288, Horst Eckart, demonstrates in this 3 – D graph the precise condition for low energy nuclear reaction through standard quantum tunnelling through a barrier of width 2a. He shows that when a approaches zero (very thin barrier), the transmission coefficient approaches 100% even when the energy of the incoming particle approaches zero. It quantum tunnels straight through a barrier represented by V sub 0 = 10. This is not possible in classical physics. Contemporary supercomputer code libraries may have programs that could compute this process for two real atoms fusing with each other, i.e. one of them quantum tunnels straight in to the other AT LOW ENERGY. This is of course the simplest first theory of quantum tunnelling, but it gives all the essentials of even the most sophisticated code. The final step for UFT228 is to incorporate relativistic effects.

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Exact Condition for LENR: Transmission plots, E dependence

Feed: Dr. Myron Evans
Posted on: Monday, September 24, 2012 11:36 PM
Author: metric345
Subject: Exact Condition for LENR: Transmission plots, E dependence

Co author of UFT228 (in prep.), Horst Eckart, demonstrates in this 3 – D graph the precise condition for low energy nuclear reaction through standard quantum tunnelling through a barrier of width 2a. He shows that when a approaches zero (very thin barrier), the transmission coefficient approaches 100% even when the energy of the incoming particle approaches zero. It quantum tunnels straight through a barrier represented by V sub 0 = 10. This is not possible in classical physics. Contemporary supercomputer code libraries may have programs that could compute this process for two real atoms fusing with each other, i.e. one of them quantum tunnels straight in to the other AT LOW ENERGY. This is of course the simplest first theory of quantum tunnelling, but it gives all the essentials of even the most sophisticated code. The final step for UFT228 is to incorporate relativistic effects.

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LENR, Further Comments on Quantum Tunnelling (Note 228(6))

Feed: Dr. Myron Evans
Posted on: Saturday, September 22, 2012 10:51 AM
Author: metric345
Subject: LENR, Further Comments on Quantum Tunnelling (Note 228(6))

The results show that:

1) When V = E, x = 0 for all a, and T = 1, there is complete transmission through the barrier.
2) When a approaches zero for finite kappa, then T approaches 1, and there is complete transmission.
3) When kappa approaches zero for finite a, then T approaches 1 and complete transmission.
4) The rate of change of T with x = kappa a is minimized at x = 0.25.

When there is complete transmission, an incoming matter wave of an atom passes right through the barrier (another atom) . Under this condition, fusion has occured in one sense. These results can be refined greatly in several directions in future work. For example the relativistci theory of note 228(7) can be developed. Also, when

dT / dx = 0

there is a maximum, minimum or inflection of T in general. There may be more than one solution to this equation. At x = 0 there is a maximum of T.

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Checking my Hand Calculations in 228(6)

Feed: Dr. Myron Evans
Posted on: Saturday, September 22, 2012 8:02 AM
Author: metric345
Subject: Checking my Hand Calculations in 228(6)

Thanks again to Horst for checking eq. (20) of note 228(6) by computer algebra, the equation is correct. So quantum tunnelling is a far more complicated problem than people might think. It needs computer algebra to find its most basic features.

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LENR, Graphs of the Transmission Coefficient of Standard Quantum Tunnelling

Feed: Dr. Myron Evans
Posted on: Saturday, September 22, 2012 7:58 AM
Author: metric345
Subject: LENR, Graphs of the Transmission Coefficient of Standard Quantum Tunnelling

Many thanks to co author Horst Eckardt for this graph, which shows that the transmission coefficient of standard quantum tunnelling theory is a maximum of one at

x = kappa a = 0

and falls to zero monotonically with increasing x. The function dT / dx goes through a minimum at

x = 0.25

The dT / dx function is exceedingly complicated and has to be evaluated by computer algebra, which also evaluates the complex number algebra. So to maximize the chances of low energy nuclear reactions, the transmission coefficient must be maximized, so kappa must be very low for a given a (thickness of the square well), or a must be very small (thin sample) for a given kappa (the wavenumber inside the square well). It could be that the fused entity is at its most stable condition at the minimum of dT / dx, the point at which the transmission coefficient changes the most slowly with x = kappa a. It was found that Merzbacher’s calculations are correct, (E. Merzbacher, “Quantum Mechanics”, Wiley, second edition), but those of P. W. Atkins for the same problem are wildly erroneous (P. W. Atkins, “Moelcular Quantum Mechanics”, Oxford University Press) due to erroneous alegbra and maybe more errors in concept. The next step is to input spacetime resonant absorption now that the baseline problem has been defined.

228(6).pdf

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228(7) : LENR, Relativistic Theory of Quantum Tunnelling

Feed: Dr. Myron Evans
Posted on: Saturday, September 22, 2012 7:38 AM
Author: metric345
Subject: 228(7) : LENR, Relativistic Theory of Quantum Tunnelling

In this case the trasnmission coefficient is given by eq. (37), with kappa and k defined by eqns. (36) and (33). The new relativistic generalization of the Schroedinger equation is eq. (34).

a228thpapernotes7.pdf

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