High Energy D 2 Bond from Feynman s Integral Wave Equation

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1 Applying the Scientific Method to Understanding Anomalous Heat Effects: Opportunities and Challenges High Energy D Bond from Feynman s Integral Wave Equation By: Thomas Barnard Sponsored by: Coolesence Special Thanks to: Mat McConnell Rick Cantwell Jim Barnard

2 Molecular Deuterium D D e Behavior Determined by Electro-Static Forces 1

3 Molecular Deuterium D D e Behavior Determined by Electro-Static Forces and Described by Quantum Mechanics 1

4 Molecular Deuterium All Properties predicted by the Schrodinger Equation D D e 1

5 Molecular Deuterium All Properties predicted by the Schrodinger Equation D Modeled with ~Atomic Wave Functions e D e Bond is due to Exchange Energy (Non-Classical) 1

6 Molecular Deuterium All Properties predicted by the Schrodinger Equation D Modeled with ~Atomic Wave Functions e D e Bond is due to Exchange Energy (Non-Classical) Born-Oppenheimer Approximation Nuclei are held Stationary while calculating the Electron Wave Function All Chemistry is Based on this!! Good Place to Start? 1

7 Molecular Deuterium All Properties predicted by the Schrodinger Equation D Modeled with ~Atomic Wave Functions e D e Fleischman-Pons affect would seem to require High K.E. Deuterons (not stationary): 1. To Localize Uncertainty Principle x >. To Overcome Coulomb Repulsion Ћ {me k } ½ 3. Similar Deuteron - Electron Energies Coupling Can not use Born-Oppenheimer Approximation 1

8 Paul Dirac s, The Lagrangian in Quantum Mechanics : 193 ( The paper that inspired Feynman s Path Integral Approach ~194) Now there is an alternative formulation for classical dynamics provided by the Lagrangian there are reasons for believing that the Lagrangian one is the more fundamental... Paul Dirac

[Paul Dirac, The Lagrangian in Quantum Mechanics : 193] the Lagrangian method allows one to collect together all the equations of motion and express them as the stationary property of a certain

9 [Paul Dirac, The Lagrangian in Quantum Mechanics : 193] the Lagrangian method allows one to collect together all the equations of motion and express them as the stationary property of a certain action function. (This action function is just the time integral of the Lagrangian). There is no corresponding action principle in terms of the coordinates and momenta of the Hamiltonian theory Paul Dirac 3

10 What Does That Mean? The Action integral accounts for particle behavior over the time, or period of an Observation The Differential Hamiltonian (Schrodinger Eq.) can not easily model correlated interference affects over periods of time and space i.e. Should not use Born-Oppenheimer with High K.E. Deuterons + Electrons 4

11 Could there be Another Form of D? Hypothesis: A correlated state of Deuterons and an Electron: the Linear Momentum State Linear Momentum State D *? D e D A Dynamic Correlation obscured by the Born-Oppenheimer Approximation 5

12 Could there be Another Form of D? Hypothesis: A correlated state of Deuterons and an Electron: the Linear Momentum State D *? A Dynamic Correlation obscured by the Born-Oppenheimer Approximation 5

13 Could there be Another Form of D? Hypothesis: A correlated state of Deuterons and an Electron: the Linear Momentum State D *? A Dynamic Correlation obscured by the Born-Oppenheimer Approximation 5

14 Could there be Another Form of D? Hypothesis: A correlated state of Deuterons and an Electron: the Linear Momentum State D *? A Dynamic Correlation obscured by the Born-Oppenheimer Approximation 5

15 If the Linear Momentum State Does Exist, it Leads to the 3 Miracles: 1- Coulomb Barrier Penetration (~30 kev bond ) - No Neutrons (coupled Electron - Nuclei) Text Book Time-Dependent Coupled Two-State Problem Also ppm tritium to 4 He ratio 3- No Gammas (coupled Electron - Nuclei) Also orders of magnitude effective bonding: ~Angstrom to Fermi Chemical Mechanism 6

16 If the Linear Momentum State Does Exist, it Leads to the 3 Miracles: 1- Coulomb Barrier Penetration (~30 kev bond ) - No Neutrons (coupled Electron - Nuclei) Text Book Time-Dependent Coupled Two-State Problem Also ppm tritium to 4 He ratio 3- No Gammas (coupled Electron - Nuclei) Also orders of magnitude effective bonding: ~Angstrom to Fermi Chemical Mechanism Nagging Question Can the Linear Momentum State be derived from accepted Q.M. principles??!! 6

17 Conceptual Understanding of the Linear Momentum State We Need 3 Basic Concepts : 1) The Action from the Lagrangian ) Path Amplitude Summation 3) Stationary State Requirements Outline D * as a Classical Bond Review Double Slit Interference Dirac s Path Amplitude Principle of least Action Summing Complex Amplitudes Stationary State Requirement Conclusion: Semi-Classical Linear Momentum State 7

18 D * Classical Bond Simulation -Radial Position vs. Time- 3 coupled F = m. a = m d x nd order Differential Eq. using dt Initial velocity: V(t=0) = Radial Distance - Fermi electron time second deuterons Radial Distance - Fermi time second deuterons 8

19 Electron acting like a Force Carrying Particle familiar to nuclear physics Classical Orbit of D * Dimensions determined by Q.M. just like the Hydrogen atom y( ) e D D 9

20 Double Slit Experiment with Electrons Feynman and Hibbs, Quantum Mechanics and Path Integrals pg. 3 Slit detector Electron source Slit 10

21 Double Slit Experiment with Electrons Feynman and Hibbs, Quantum Mechanics and Path Integrals pg. 3 Slit detector Electron source Slit 10

$4 Interference Pattern Diffraction Patterns Summed Diffraction Patterns P(x) with both Holes open P 1 (x) with just Hole 1 open P (x) with just$

22 Probability Distributions from Double Slit Experiments Feynman and Hibbs, Quantum Mechanics and Path Integrals pg. 4 Interference Pattern Diffraction Patterns Summed Diffraction Patterns P(x) with both Holes open P 1 (x) with just Hole 1 open P (x) with just Hole open P 1 (x) +P (x) P(x) 11

23 Dirac s Contact Transformation The Lagrangian in Quantum Mechanics : 193 Ø j [b,a] ~ e i ћ S j [b,a] Where: S = Action for path j (from a to b ) = the time integral of Lagrangian t b The amplitude for a single path j S = L dt Action t a L = (K.E.) - (P.E.) Lagrangian: Classical Newtonian Mechanics Classical Mechanics meets Wave Mechanics! 1

24 Dirac s Contact Transformation The Lagrangian in Quantum Mechanics : 193 Ø j [b,a] ~ e i ћ S j [b,a] Where: S = Action for path j (from a to b ) = the time integral of Lagrangian t b The amplitude for a single path j S = L dt Action t a L = (K.E.) - (P.E.) Lagrangian: Classical Newtonian Mechanics Classical Mechanics meets Wave Mechanics! 1

25 Dirac s Contact Transformation The Lagrangian in Quantum Mechanics : 193 Ø j [b,a] ~ e i ћ S j [b,a] Where: S = Action for path j (from a to b ) = the time integral of Lagrangian t b The amplitude for a single path j S = L dt Action t a L = (K.E.) - (P.E.) Lagrangian: Classical Newtonian Mechanics Classical Mechanics meets Wave Mechanics! 1

Analogous Experiment in Wave Interference Amplitude Squared = Probability S Iø 1 +ø I Amplitude waves - NOT Reala mathematical device Ø j [b,a] ~ e i ћ S j = cos(s j /ћ)+

26 Analogous Experiment in Wave Interference Amplitude Squared = Probability S Iø 1 +ø I Amplitude waves - NOT Reala mathematical device Ø j [b,a] ~ e i ћ S j = cos(s j /ћ)+ i sin(s j /ћ) Feynman and Hibbs, Quantum Mechanics and Path Integrals pg. 5 Euler s Formula The easiest way to represent wave amplitudes is by complex numbers... -Feynman 13

27 14 Feynman s Wave Equation Can Derive Schrodinger Equation from this Concept Sum over all possible paths in time and space i S ћ Ψ[b(t b )] = {e Ψ[a(t a )]} Only paths Feynman Recall: t b S = L dt Action t a L = (K.E.) - (P.E.)

28 14 Feynman s Wave Equation Can Derive Schrodinger Equation from this Concept Sum over all possible paths in time and space i S ћ Ψ[b(t b )] = {e Ψ[a(t a )]} Now think infinite Slits Feynman Recall: t b Can be solved but Conceptual Approach today S = L dt Action t a L = (K.E.) - (P.E.)

29 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} t b S = Action = t a m v V(x,y,z) dt 15

30 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} S = Action = t b m v V(x,y,z) dt 0 for free particle t a Action of a Free particle x a t a a b x b t b Classical Path From a to b (constant velocity) 15

31 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} S = Action = t b t a m v V(x,y,z) dt Action of a Free particle a b x a t a x b t b S cl = m x t ( t) Classical Action 15

32 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} S = Action = t b t a m v V(x,y,z) dt Action of a Free particle x a t a a y b x b t b S cl = m x t ( t) non-classical Paths: think Double Slit 15

33 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} t b S = Action = m v V(x,y,z) dt t a Action of a Free particle a y b x a t a x b t b Recall: S cl = m x t ( t) non-classical Paths: think Double Slit to Infinite Slits a b 15

34 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} S = Action = t b t a m v V(x,y,z) dt Action of a Free particle x a t a a y b x b t b S cl = m x t ( t) Classical Action y = 0 S y = m x t + y t ( t) Action of Alternate Paths (non-classical Action) 15

35 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} S = Action = t b t a m v V(x,y,z) dt Action of a Free particle x a t a a y b x b t b S S cl = m x t ( t) Classical Action y = 0 y S y = m x t + y t ( t) Action of Alternate Paths (non-classical Action) 15

36 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} S = Action = t b t a m v V(x,y,z) dt Action of a Free particle x a t a a y b x b t b S S cl = m x t ( t) y S y = m x t + y t ( t) Action of Alternate Paths (non-classical Action) 15

37 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} t b S = Action = m v V(x,y,z) dt t a Action of a Free particle x a t a a y b x b t b S S cl = m x t ( t) Classical Path Extremum i.e. ds = 0 = ma - F dy y S y = m x t + y t ( t) Newtons s nd Law Least Action is Classical Path 15

38 The Principle of Least Action The Lagrangian = {K.E.} {P.E.} t b S = Action = t a m v V(x,y,z) dt ` Recall: D * Classical Motion x a t a a Action of a Free particle y b x b t b An Extremum S S cl = m x t ( t) Classical Path Extremum i.e. ds = 0 = ma - F dy y S y = m x t + y t ( t) Newtons s nd Law Least Action is Classical Path 15

39 Summing Amplitudes of Paths Free Particle Examples S = Action = t b t a m v[x,t] + v[y,t] dt y( t, 0) y( t, 15) y( t, 49) y( t, 50) Y= Distance from Classical Path y k =15 k =49 k = a x( t) X Distance b Classical free particle path 16

40 Summing Amplitudes of Paths Free Particle Examples S = Action = t b t a m v[x,t] + v[y,t] dt y( t, 0) y( t, 15) y( t, 49) y( t, 50) Y= Distance from Classical Path y y*15 y*14 k =15 y*5 k =49 y k =50 Non-Classical paths 1 to a x( t) X Distance b Classical free particle path 16

41 Example # 1: Electron with 1 meter path at ~13.6 ev S j /ћ = action / ћ Note the very large action/ћ ~ y Sum of Paths 15 Total Amplitude= e is(j)/ћ J= meters from Classical Path Action in units of Planks Constant 17

42 Example # 1: Electron with 1 meter path at ~13.6 ev S j /ћ = action / ћ Note the very large action/ћ ~ y Sum of Paths 15 Total Amplitude= e is(j)/ћ J=0 15 = [cos(s j /ћ) + i sin(s j /ћ)] J= meters from Classical Path Like adding Vectors in the Complex Plane imaginary 17

43 Example # 1: Electron with 1 meter path at ~13.6 ev S j /ћ = action / ћ Note the very large action/ћ ~ y Sum of Paths 15 Total Amplitude= e is(j)/ћ J=0 15 = [cos(s j /ћ) + i sin(s j /ћ)] J=0 x= Real y= imaginary meters from Classical Path Complex Plane Imaginary 1 7 sy ( k ) Real sx ( k )

44 Example # 1: Electron with 1 meter path at ~13.6 ev S j /ћ = action / ћ Note the very large action/ћ ~ y Sum of Paths 15 Total Amplitude= e is(j)/ћ J=0 15 = [cos(s j /ћ) + i sin(s j /ћ)] J=0 x= Real y= imaginary meters from Classical Path Complex Plane Paths near Extremum Contribute to Sum of amplitudes (i.e. Observables ) Imaginary 1 7 sy ( k ) Real sx ( k )

45 Example # 1: Electron with 1 meter path at ~13.6 ev S j /ћ = action / ћ Note the very large action/ћ ~ y Sum of Paths 15 Total Amplitude= e is(j)/ћ J=0 15 = [cos(s j /ћ) + i sin(s j /ћ)] J=0 x= Real y= imaginary meters from Classical Path Complex Plane Paths near Extremum Contribute to Sum of amplitudes (i.e. Observables ) Imaginary 1 7 sy ( k ) Real sx ( k )

46 Example # 1: Electron with 1 meter path at ~13.6 ev S j /ћ = action / ћ Note the very large action/ћ ~ y Sum of Paths 15 Total Amplitude= e is(j)/ћ J=0 15 = [cos(s j /ћ) + i sin(s j /ћ)] J=0 x= Real y= imaginary meters from Classical Path Complex Plane Remember: Large Action = Small Deviation from Classical Path Resultant deviation from Classical path ~ 4.5 microns Imaginary 1 7 sy( k) Real sx( k)

47 15 Example # : Electron with 1 Angstrom Path at ~13.6 ev much smaller action/ћ ~ 1 Hydrogen Electron K.E. 10 action / ћ meters from Classical Path Sum of Paths Complex Plane 6 Imaginary sy( k) Deviation from Classical path ~ 0.45 Angstroms (diameter ~0.9 Angstrom) Real sx( k) 18

48 15 Example # : Electron with 1 Angstrom Path at ~13.6 ev much smaller action/ћ ~ 1 Hydrogen Electron K.E. 10 action / ћ meters from Classical Path Sum of Paths Complex Plane 6 Paths real and maximum non-real 5 0) 14) t, 14) a Classical path Non-real path b Circular Electron Orbit with d ~1Ǻ Imaginary Deviation from Classical path ~ 0.45 Angstroms (diameter ~0.9 Angstrom) sy( k) Real sx( k) 18

49 15 Example # : Electron with 1 Angstrom Path at ~13.6 ev much smaller action/ћ ~ 1 Hydrogen Electron K.E. action / ћ 10 5 Origin of Atomic Dimensions -Determined by Plank s Constant meters from Classical Path Sum of Paths Complex Plane 6 Paths real and maximum non-real 5 0) 14) t, 14) a Classical path Non-real path b Circular Electron Orbit with d ~1Ǻ Imaginary Deviation from Classical path ~ 0.45 Angstroms (diameter ~0.9 Angstrom) sy( k) Real sx( k) 18

50 15 Example # : Electron with 1 Angstrom Path at ~13.6 ev much smaller action/ћ ~ 1 Hydrogen Electron K.E. action / ћ 10 5 Origin of Atomic Dimensions -Determined by Plank s Constant- 0) 14) t, 14) a Small Action ill-defined Path meters from Classical Path Large Action Classical Path Paths real and maximum non-real Classical path Non-real path b Circular Electron Orbit with d ~1Ǻ Remember: Imaginary Deviation from Classical path ~ 0.45 Angstroms (diameter ~0.9 Angstrom) sy( k) Sum of Paths Complex Plane Real sx( k) 18

51 Stationary State Requirement Repeating Amplitude of Path a b e Classical orbit of Electron in Deuterium atom D 19

52 Stationary State Requirement Repeating Amplitude of Path a a b e b Classical orbit of Electron in Deuterium atom D 19

53 Stationary State Requirement Repeating Amplitude of Path a a b e Classical orbit of Electron in Deuterium atom b D 19

54 Stationary State Requirement Repeating Amplitude of Path a b e Classical orbit of Electron in Deuterium atom D Amplitude along path a path made linear to Make the Amplitude easier to graph b 19

55 Stationary State Requirement Repeating Amplitude of Path Stationary State is like Free Particle amplitude repeating Exact Path (Classical or Non-Classical Paths) a D b e Classical orbit of Electron in Deuterium atom Amplitude along path a path made linear to Make the Amplitude easier to graph b 19

56 Stationary State Requirement Repeating Amplitude of Path c a b y( ) Dramatized Classical orbit of electron in D * molecule a b Deuterons Classical Orbit 0

57 Stationary State Requirement Repeating Amplitude of Path c a b y( ) Classical orbit of electron in D * molecule a c 0

58 Stationary State Requirement Repeating Amplitude of Path c D a b y( ) Classical orbit of electron in D * molecule With Born-Oppenheimer a b If we invoke the Linear Momentum State with Born-Oppenheimer D 0

59 Stationary State Requirement Repeating Amplitude of Path Recall: Example # Paths real and maximum non-real c D a b y( ) Classical orbit of electron in D * molecule With Born-Oppenheimer a b a Classical path Non-real path b x( t) D Linear Momentum State with Born-Oppenheimer Get other paths with similar probability Results in Standard Schrodinger Solution, (i.e. Normal D ) 0

60 Stationary State Requirement Repeating Amplitude of Path c a b y( ) Classical orbit of electron in D * molecule a c Amplitude along path a path made linear to Make the Amplitude easier to graph (not to scale) c 0

61 Stationary State Requirement Repeating Amplitude of Path c a b y( ) Classical orbit of electron in D * molecule a c Amplitude along path a path made linear to Make the Amplitude easier to graph (not to scale) c 0

62 Stationary State Requirement Repeating Amplitude of Path c a b y( ) Classical orbit of electron in D * molecule a b Amplitude along path a c Expand to full Orbit : a b path made linear to Make the Amplitude easier to graph (not to scale) b 0

63 Stationary State Requirement Repeating Amplitude of Path Recall that large action Classical like Path c a b y( ) Classical orbit of electron in D * molecule a b Very Long Path Gives Very Large Action Crosses axis ~400 times Amplitude along path a c path made linear to Make the Amplitude easier to graph (not to scale) b 0

64 n~ 400 is like a very large quantum no.= very large Action + Action is also Very Sensitive to Large Potential through Nuclei Conceptual Conclusion Classical Like Electron Behavior The Born-Oppenheimer Approximation obscures this straight forward Quantum Solution The Schrodinger Equation Can Not Easily Describe this Correlation The Correlation is Accessible through Feynman s Wave Equation over an Orbital Period The Q.M. Correlation is defined by an Extremum of the Lagrangian Action of a Classically defined orbit Yes! - The Linear Momentum State can be derived from Basic Accepted Q.M. Principles 1

Conclusion Classical Like Electron Behavior D D 400 Fm Linear Momentum State Electron Probability Relative

65 n~ 400 is like a very large quantum no.= very large Action + Action is also Very Sensitive to Large Potential through Nuclei Conceptual Conclusion Classical Like Electron Behavior D D 400 Fm Linear Momentum State Electron Probability Relative to D s from Feynman s Wave Equation Ǻ Electron Probability Y Ǻ Top view note Order of Magnitude difference in the scales [axially symmetric] Y D D Side view emission ene

66 Questions? Evolution Valley California Sierras The 4 th Miracle

67 END of PRESENTATION SLIDES

68 Pseudo Potential and Resulting Ground State Wave Function for D * Calculated numerically with the Schrodinger equation Pseudo Potential for Deuterons = ev Deuterium Separation Fermi (=10-15 m) Reduced mass wave function of D * with -33,000 ev Pseudo Potential 33,000 ev

69 Pseudo Potential and Resulting Excited State Wave Function for D * Calculated numerically with the Schrodinger equation Pseudo Potential for deuterons - ev Wave function at ~ -130 ev state Deuterium Separation Fermi (=10-15 m) Pseudo Potential

70 Electron Energy as a Function of Deuterium Separation Deuteron Separation R Fermi (=10-15 m) Electron Energy - ev R=5000 Sample electron Wave functions at various D separation R= R=80

71 Approximations Used in Calculation 1. Nuclear Charge distribution and Relativistic affect. Less accurate electron energy calculation. No affect on over all theory, with moderate affect on numbers.. Classical path assumed to go through center of mass. Very small error in electron Action. Not much affect on results.

72 = maximum deviation of electron path from the center of potential between the two deuterons Axis through nuclei at t 6 t 0 t 1 t t 3 t 4 t 5 t6 t7 t8 D1 Electron path t 9 t 10 t11 <1F t 11 t 10 t 9 t 8 t 7 D t 6 t 5 t 4 t 3 t t 1 t 0

73 Origin of Maximum Action for Linear Momentum State from Circular Orbit Least Action r 3 R r 4 r 0 r r 1 r r 3 S ћ r r 1 R r 4 Linear Momentum State solution Hydrogen atom solution

74 Stationary State Requirement Repeating Amplitude of Path t 5 t 4 c D D a b t 1 t 0 y( ) Classical orbit of electron in D * molecule a c D D t t 3 0

75 Lagrangian the L x x x L x x S where A dx A dx x x S i Lim t x i i i i i x x x x x x k k k i i i k k k k k k k = = = Ψ + + = +, ), ( : ), ( exp ), ( ε ε ε Need to Solve Feynman s Wave Equation This is doable, but we will take an intuitive approach today ICCF 18 July 1-7, = Classical Newtonian Mechanics Feynman Can Derive Schrodinger Equation from this Equation

76 z Classical Newtonian Simulation of 3 Particle Bond δ electron F 3 =m 3 * a 3 = d{v 1,3 +V,3 } dz Initial conditions R -R D F =m * a = d{v 1, +V,3 } dz v 1 = v = v 3 = 0 x 1 = -R x = R x 3 = R + δ D1 F 1 =m 1 * a 1 = d{v 1, +V 1,3 } dz Note: The proton distribution in the nucleus is used to calculated the charge distribution in the nucleus 6

Exp. 4. Quantum Chemical calculation: The potential energy curves and the orbitals of H2 +

Exp. 4. Quantum Chemical calculation: The potential energy curves and the orbitals of H2 + 1. Objectives Quantum chemical solvers are used to obtain the energy and the orbitals of the simplest molecules