LOW ENERGY NUCLEAR FUSION REACTIONS: QUANTUM TUNNELLING

Transcription

1 Journal of Foundations of Physics and Chemistry 13 (1) LOW NRGY NUCLAR FUSION RACTIONS: QUANTUM TUNNLLING M. W. vans, H. ckardt and D. W. Lindström Alha Institute for Advanced Studies ( ( A new linear equation is develoed of relativistic quantum mechanics and the equation is alied to the theory of quantum tunnelling based on the Schrödinger equation in the non relativistic limit. Using a square barrier model in the first aroximation, it is shown that low energy nuclear fusion occurs as a result of the Schrödinger equation, which is a limit of the C fermion equation. It is shown that for a thin samle and a given barrier height, 1% transmission occurs by quantum tunnelling even when the energy of the incoming article aroaches zero. This is therefore a lausible model of low energy nuclear reaction. The new relativistic equation is used to study relativistic corrections. Absortion of quanta of sacetime may result in enhancement of the quantum tunnelling rocess. Keywords: Limits of C theory, linear relativistic quantum mechanics, low energy nuclear reaction, quantum tunnelling 1. Introduction Recently in this series of aers [1 1] the C fermion equation has been used to give an exlanation of low energy nuclear reaction (LNR [11]), which has been observed exerimentally to be reroducible and reeatable, and which has been develoed into a new source of energy. In this aer the lausibility of LNR is examined with a new linear tye of relativistic quantum mechanics which can be derived straightforwardly from classical secial relativity, a well defined limit of C theory. It is shown in Section that the instein energy equation can be quantized directly into a new tye of linear, relativistic Schrödinger equation which reduces in the non

2 M. W. vans, H. ckardt and D. W. Lindstrom relativistic limit to the Schrödinger equation. It is well known [1] that the latter is the basis for quantum tunnelling theory, and can be solved to give the transmission coefficient of quantum tunnelling. In Section, the well known standard theory [1] of quantum tunnelling is used with a rectangular barrier of thickness a and height V where V is the otential energy. This is a simle but instructive model of nuclear fusion in which an incoming atom meets the Coulomb barrier of a second atom and tunnels into it, causing nuclear fusion. It is shown that 1% transmission (comlete tunnelling) can occur for a thin samle when the energy of the incoming article aroaches zero for a finite V. This rocess is grahed in Section 3. Relativistic corrections of this simle theory can be develoed from the new linear equation derived in this aer of relativistic quantum mechanics. Relativistic corrections are grahed and discussed in Section 3.. Linear equation for relativistic quantum mechanics and alication to the transition coefficient of quantum tunelling Consider the instein equation of secial relativity [1]: where is the total relativistic energy: 4 = c + m c (1) =γmc () and where is the relativistic momentum: =γm v (3) Here m is the article mass, c the seed of light in vacuo, and γ is the Lorentz factor 1 / v γ= 1 c (4) where v is the article velocity. The classical relativistic hamiltonian is H = γmc + V (5) where V is the otential energy. The roblem faced by the ioneers of relativistic quantum mechanics was the quantization of q. (1) using the Schrödinger ostulate: = i (6) µ µ i.e.: ˆ = i, ˆ = i t v (7)

3 q. (1) roduced the Klein Gordon equation: Low nergy Nuclear Fusion Reactions 3 + mc ψ = (8) ħ and the Dirac equation, which has been recently develoed into the C fermion equation in UFT17 ff. of this series. These are all non-linear in because of the structure of q. (1). Consider q. (1) in the format: By using the momentum oerator: 1 = γmc = + mc γm (9) ˆ = i v (1) q. (9) becomes a linear, relativistic Schrödinger equation of a new tye Ĥψ= ψ (11) where the relativistic hamiltonian eigenoerator for a free article is: and where the total energy eigenvalues are: ˆ 1 H= + mc γm ( ˆ ) (1) =γmc (13) The eigenfunction ψ is the wave function of the Schrödinger equation generalized to relativistic quantum mechanics. It follows that: ˆ v mc 1 ψ= ψ= γ ψ (14) The Schrödinger ostulate (6) combined with the de Broglie instein ostulate is: where: = i = (15) µ µ µ 1 ω =,, =,, =, c c t v c (16) µ µ µ Here ω is the frequency of the matter wave, and is its wavenumber. Therefore: ˆ ψ= ψ= mc ( γ 1) ψ= mc ψ c (17) For a free wave / article: mc 1 = ( γ 1 ) / (18)

4 4 M. W. vans, H. ckardt and D. W. Lindstrom and in the non-relativistic limit: mc 1 v ( γ 1)= mc 1 1 mv c vc (19) mv () which is the classical relation between momentum and velocity. For the uroses of quantum tunnelling theory denote: and: k = mc ( γ 1 1 ) / (1) In the resence of otential energy V the oerator (1) becomes: 1 Hˆ = ( ˆ + mc ) + V γm In quantum tunnelling theory we wish to consider: ( ) () ˆ ψ= ψ= γm V mc ψ (3) 1 = γ ( m( V) m c ) (4) < V (5) so we define: Denote the rest wavenumber by: then arrive at the definition: q. (1) can be written as: 1 1 ( ) = γmv / (6) k = mc γm + = V + = mc γ (7) (8) (9)

5 Low nergy Nuclear Fusion Reactions 5 where =γmc (3) so we arrive at the definition: k γm + = In order to forge a recise analogy with the Schrödinger equation write q. (14) as In the non relativistic limit: (31) ˆ mc ψ= ( γ 1) ψ (3) m 1 mc mc v 1 ( γ 1)= 1 1 mv c vc and reduces to the classical kinetic energy of a free article: (33) = T = 1 mv (34) and: which is the non relativistic limit of: k V m ψ = ψ k = m In the resence of a otential, q. (3) becomes: and: (35) (36) m mc =, = ( γ 1) (37) ˆ mc + V ψ= ψ= ( γ 1) ψ m (38) v ψ = ψ = m V ψ (39) m mc = γ 1 ( V ), = ( ) (4) V >

6 6 M. W. vans, H. ckardt and D. W. Lindstrom It is well known [1] that the transmission coefficient of quantum tunnelling is: for a otential of the tye: ( + ) T = k 8 k cos 4a k 6 k (41) V =, x< a, V = V, a < x< a, V =, x > a, (4) < V (43) in which: k = m, = mc γ 1 (44) = = mv, mc γ 1 (45) In Section 3, various results from the standard equation (41) are grahed with the intention of finding the otimal condition for low energy nuclear reaction described through the rocess of quantum tunnelling. These results are augmented by considerations based on the new relativistic Schrödinger equation (38). This is a simle first theory, contemorary suercomuters and code ackages can be alied to the roblem of simulating the fusion of one atom with another. The analysis in Section 3 shows that the single most imortant factor is the mass m of the incoming article. The extra ingredient given by C theory is the ossibility of augmenting this standard quantum tunnelling theory with resonant absortion of quanta of sacetime energy. That will be the subject of future work. 3 GRAPHICAL ANALYSIS AND DISCUSSION Section by Horst ckardt and Douglas Lindstrom TO B ADDD ACKNOWLDGMNTS The British Government is thanked for a Civil List ension and the staff of AIAS and others for many interesting discussions. Dave Burleigh is thanked for voluntary osting and Alex Hill, Robert Cheshire and Simon Clifford for translation and broadcasting. The AIAS is governed by the Newlands Family Trust established 1.

7 Low nergy Nuclear Fusion Reactions 7 RFRNCS [1] M. W. vans, d., Journal of Foundations of Physics and Chemistry, (Cambridge International Science Publishing, CISP, 11 onwards). [] M. W. vans, S. J. Crothers, H. ckardt and K. Pendergast, Criticisms of the instein Field quation (CISP, 11). [3] M. W. vans, d., Definitive Refutations of the insteinian General Relativity (CISP, 1, Secial Issue Six of ref. (1)). [4] M. W. vans, H. ckardt and D. W. Lindstrom, Generally Covariant Unified Field Theory (Abramis Academic, 5-11) in seven volumes. [5] M. W. vans, H. ckardt and D. W. Lindstrom, aers and lenary in the journals of the Serbian Academy of Sciences. [6] M. W. vans and S. Kielich, ds., Modern Nonlinear Otics (Wiley, 199, 1993, 1997, 1) in two editions and six volumes. [7] M. W. vans and L. B. Crowell, Classical and Quantum lectrodynamics and the B(3) Field (World Scientific, 1); [8] M. W. vans and A. A. Hasanein, The Photomagneton in Quantum Field Theory (World Scientific 1994). [9] M. W. vans and J.- P. Vigier, The nigmatic Photon (Kluwer, 1994 to ) in ten volumes hardback and softback. [1] K. Pendergast, The Life of Myron vans (CISP, 11). [11] LNR Conference, University of Zurich, Set. 1 (see diary or blog of [1]. Merzbacher, Quantum Mechanics (Wiley, nd. ed., 197),. 191 ff.