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Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Will Hydrinos Save the World? An evaluation of Randell Mills’ Classical Quantum Mechanics by a skeptical mathematician Willie W. Wong Department of Mathematics Princeton University September 16, 2006 meeting of the Philadelphia Association for Critical Thinking Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Outline 1 Introduction: What is CQM A bit of history Overview of the claims Overview: what this talk is and isn’t about. 2 The Theory: Does it Work? Problem 1: The Classical Wave Equation Problem 2: The δ Function Solution Problem 3: The Haus Condition 3 Some other objections Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Outline 1 Introduction: What is CQM A bit of history Overview of the claims Overview: what this talk is and isn’t about. 2 The Theory: Does it Work? Problem 1: The Classical Wave Equation Problem 2: The δ Function Solution Problem 3: The Haus Condition 3 Some other objections Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Outline 1 Introduction: What is CQM A bit of history Overview of the claims Overview: what this talk is and isn’t about. 2 The Theory: Does it Work? Problem 1: The Classical Wave Equation Problem 2: The δ Function Solution Problem 3: The Haus Condition 3 Some other objections Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What is it and why do we care? Since the industrial revolution there’s a large market for “energy”. From the steam engine, ICE, to modern electricity, we have a need to tap into natural energy reserves. So energy research = $$$ The unobtainable holy grail, energy conservation, magnetics. Slightly more mainstream: ZPE, cold fusion. Experiments are not accessible to layman. BLP + CQM: experiment and theory. Accessible to armchair scientists. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What is it and why do we care? Since the industrial revolution there’s a large market for “energy”. From the steam engine, ICE, to modern electricity, we have a need to tap into natural energy reserves. So energy research = $$$ The unobtainable holy grail, energy conservation, magnetics. Slightly more mainstream: ZPE, cold fusion. Experiments are not accessible to layman. BLP + CQM: experiment and theory. Accessible to armchair scientists. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What is it and why do we care? Since the industrial revolution there’s a large market for “energy”. From the steam engine, ICE, to modern electricity, we have a need to tap into natural energy reserves. So energy research = $$$ The unobtainable holy grail, energy conservation, magnetics. Slightly more mainstream: ZPE, cold fusion. Experiments are not accessible to layman. BLP + CQM: experiment and theory. Accessible to armchair scientists. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What is it and why do we care? Since the industrial revolution there’s a large market for “energy”. From the steam engine, ICE, to modern electricity, we have a need to tap into natural energy reserves. So energy research = $$$ The unobtainable holy grail, energy conservation, magnetics. Slightly more mainstream: ZPE, cold fusion. Experiments are not accessible to layman. BLP + CQM: experiment and theory. Accessible to armchair scientists. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What is it and why do we care? Since the industrial revolution there’s a large market for “energy”. From the steam engine, ICE, to modern electricity, we have a need to tap into natural energy reserves. So energy research = $$$ The unobtainable holy grail, energy conservation, magnetics. Slightly more mainstream: ZPE, cold fusion. Experiments are not accessible to layman. BLP + CQM: experiment and theory. Accessible to armchair scientists. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary BlackLight Power and Dr. Randell Mills Randell Mills, MD B.A. chemistry summa cum laude and Phi Betta Kappa from Franklin and Marshall ’82 M.D. Harvard Medical School ’86 One year of graduate work in electrical engineering at MIT, where he read the work of H.A. Haus BlackLight Power, Inc. of Cranbury, NJ Founded by Randell Mills, the current president, director, and chairman of the board. Performs research to exploit the BlackLight Process of hydrino theory for energy generation. Claims to have over $50M in venture capital as of this year. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary BlackLight Power and Dr. Randell Mills Randell Mills, MD B.A. chemistry summa cum laude and Phi Betta Kappa from Franklin and Marshall ’82 M.D. Harvard Medical School ’86 One year of graduate work in electrical engineering at MIT, where he read the work of H.A. Haus BlackLight Power, Inc. of Cranbury, NJ Founded by Randell Mills, the current president, director, and chairman of the board. Performs research to exploit the BlackLight Process of hydrino theory for energy generation. Claims to have over $50M in venture capital as of this year. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary BlackLight Power and Dr. Randell Mills Randell Mills, MD B.A. chemistry summa cum laude and Phi Betta Kappa from Franklin and Marshall ’82 M.D. Harvard Medical School ’86 One year of graduate work in electrical engineering at MIT, where he read the work of H.A. Haus BlackLight Power, Inc. of Cranbury, NJ Founded by Randell Mills, the current president, director, and chairman of the board. Performs research to exploit the BlackLight Process of hydrino theory for energy generation. Claims to have over $50M in venture capital as of this year. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary BlackLight Power and Dr. Randell Mills Randell Mills, MD B.A. chemistry summa cum laude and Phi Betta Kappa from Franklin and Marshall ’82 M.D. Harvard Medical School ’86 One year of graduate work in electrical engineering at MIT, where he read the work of H.A. Haus BlackLight Power, Inc. of Cranbury, NJ Founded by Randell Mills, the current president, director, and chairman of the board. Performs research to exploit the BlackLight Process of hydrino theory for energy generation. Claims to have over $50M in venture capital as of this year. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary BlackLight Power and Dr. Randell Mills Randell Mills, MD B.A. chemistry summa cum laude and Phi Betta Kappa from Franklin and Marshall ’82 M.D. Harvard Medical School ’86 One year of graduate work in electrical engineering at MIT, where he read the work of H.A. Haus BlackLight Power, Inc. of Cranbury, NJ Founded by Randell Mills, the current president, director, and chairman of the board. Performs research to exploit the BlackLight Process of hydrino theory for energy generation. Claims to have over $50M in venture capital as of this year. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary BlackLight Power and Dr. Randell Mills Randell Mills, MD B.A. chemistry summa cum laude and Phi Betta Kappa from Franklin and Marshall ’82 M.D. Harvard Medical School ’86 One year of graduate work in electrical engineering at MIT, where he read the work of H.A. Haus BlackLight Power, Inc. of Cranbury, NJ Founded by Randell Mills, the current president, director, and chairman of the board. Performs research to exploit the BlackLight Process of hydrino theory for energy generation. Claims to have over $50M in venture capital as of this year. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM: A Classical Theory CQM is a classical theory that can be derived from “ﬁrst principles”. Accepts Newtonian/relativistic mechanics, Maxwell’s equations, thermodynamics, etc. Also accepts some tenets of quantum mechanics (QM): quantization of energy levels, de Broglie wavelength. Macroscopic physics and Microscopic physics should correspond. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM: A Classical Theory CQM is a classical theory that can be derived from “ﬁrst principles”. Accepts Newtonian/relativistic mechanics, Maxwell’s equations, thermodynamics, etc. Also accepts some tenets of quantum mechanics (QM): quantization of energy levels, de Broglie wavelength. Macroscopic physics and Microscopic physics should correspond. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM: A Classical Theory CQM is a classical theory that can be derived from “ﬁrst principles”. Accepts Newtonian/relativistic mechanics, Maxwell’s equations, thermodynamics, etc. Also accepts some tenets of quantum mechanics (QM): quantization of energy levels, de Broglie wavelength. Macroscopic physics and Microscopic physics should correspond. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM: A Classical Theory CQM is a classical theory that can be derived from “ﬁrst principles”. Accepts Newtonian/relativistic mechanics, Maxwell’s equations, thermodynamics, etc. Also accepts some tenets of quantum mechanics (QM): quantization of energy levels, de Broglie wavelength. Macroscopic physics and Microscopic physics should correspond. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM’s (alleged) Advantages Over QM Lorentz-invariant Deterministic Gives exact solutions, rather than approximations Explains non-radiation of electrons in orbit Agrees with current observations Is self-consistent (?!) ... and the list goes on. c.f. the “Introduction” of Randall Mills’ The Grand Uniﬁed Theory of Classical Quantum Mechanics for a list of objections of Schrödinger’s wave mechanics that underlies QM. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM’s (alleged) Advantages Over QM Lorentz-invariant Deterministic Gives exact solutions, rather than approximations Explains non-radiation of electrons in orbit Agrees with current observations Is self-consistent (?!) ... and the list goes on. c.f. the “Introduction” of Randall Mills’ The Grand Uniﬁed Theory of Classical Quantum Mechanics for a list of objections of Schrödinger’s wave mechanics that underlies QM. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM’s (alleged) Advantages Over QM Lorentz-invariant Deterministic Gives exact solutions, rather than approximations Explains non-radiation of electrons in orbit Agrees with current observations Is self-consistent (?!) ... and the list goes on. c.f. the “Introduction” of Randall Mills’ The Grand Uniﬁed Theory of Classical Quantum Mechanics for a list of objections of Schrödinger’s wave mechanics that underlies QM. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM’s (alleged) Advantages Over QM Lorentz-invariant Deterministic Gives exact solutions, rather than approximations Explains non-radiation of electrons in orbit Agrees with current observations Is self-consistent (?!) ... and the list goes on. c.f. the “Introduction” of Randall Mills’ The Grand Uniﬁed Theory of Classical Quantum Mechanics for a list of objections of Schrödinger’s wave mechanics that underlies QM. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM’s (alleged) Advantages Over QM Lorentz-invariant Deterministic Gives exact solutions, rather than approximations Explains non-radiation of electrons in orbit Agrees with current observations Is self-consistent (?!) ... and the list goes on. c.f. the “Introduction” of Randall Mills’ The Grand Uniﬁed Theory of Classical Quantum Mechanics for a list of objections of Schrödinger’s wave mechanics that underlies QM. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM’s (alleged) Advantages Over QM Lorentz-invariant Deterministic Gives exact solutions, rather than approximations Explains non-radiation of electrons in orbit Agrees with current observations Is self-consistent (?!) ... and the list goes on. c.f. the “Introduction” of Randall Mills’ The Grand Uniﬁed Theory of Classical Quantum Mechanics for a list of objections of Schrödinger’s wave mechanics that underlies QM. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary CQM’s (alleged) Advantages Over QM Lorentz-invariant Deterministic Gives exact solutions, rather than approximations Explains non-radiation of electrons in orbit Agrees with current observations Is self-consistent (?!) ... and the list goes on. c.f. the “Introduction” of Randall Mills’ The Grand Uniﬁed Theory of Classical Quantum Mechanics for a list of objections of Schrödinger’s wave mechanics that underlies QM. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary Hydrino Theory There exists states with fractional principle quantum 1 numbers n where n is an integer–hydrogen atoms in such states are called hydrinos. n is limited by the fact that the electron in orbit cannot exceed the speed of light. n = 1 (QM ground state) is metastable: transition to the fractional states can only occur by non-radiative energy transfers, i.e. collisions of hydrogen atoms with catalyst. BlackLight Process: the catalyst should release the energy obtained from the hydrogen atom in the collision by photo-emission—which would be in the ultraviolet, a.k.a. “black light”, regime. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary Hydrino Theory There exists states with fractional principle quantum 1 numbers n where n is an integer–hydrogen atoms in such states are called hydrinos. n is limited by the fact that the electron in orbit cannot exceed the speed of light. n = 1 (QM ground state) is metastable: transition to the fractional states can only occur by non-radiative energy transfers, i.e. collisions of hydrogen atoms with catalyst. BlackLight Process: the catalyst should release the energy obtained from the hydrogen atom in the collision by photo-emission—which would be in the ultraviolet, a.k.a. “black light”, regime. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary Hydrino Theory There exists states with fractional principle quantum 1 numbers n where n is an integer–hydrogen atoms in such states are called hydrinos. n is limited by the fact that the electron in orbit cannot exceed the speed of light. n = 1 (QM ground state) is metastable: transition to the fractional states can only occur by non-radiative energy transfers, i.e. collisions of hydrogen atoms with catalyst. BlackLight Process: the catalyst should release the energy obtained from the hydrogen atom in the collision by photo-emission—which would be in the ultraviolet, a.k.a. “black light”, regime. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary Hydrino Theory There exists states with fractional principle quantum 1 numbers n where n is an integer–hydrogen atoms in such states are called hydrinos. n is limited by the fact that the electron in orbit cannot exceed the speed of light. n = 1 (QM ground state) is metastable: transition to the fractional states can only occur by non-radiative energy transfers, i.e. collisions of hydrogen atoms with catalyst. BlackLight Process: the catalyst should release the energy obtained from the hydrogen atom in the collision by photo-emission—which would be in the ultraviolet, a.k.a. “black light”, regime. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What I Won’t Talk About Experimental evidence and their interpretations The collision model of energy transfer The various “exact solutions” for atoms and molecules that BLP research claimed to have found The inner functionings of the orbitsphere model and the notion of the trap photon. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What I Won’t Talk About Experimental evidence and their interpretations The collision model of energy transfer The various “exact solutions” for atoms and molecules that BLP research claimed to have found The inner functionings of the orbitsphere model and the notion of the trap photon. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What I Won’t Talk About Experimental evidence and their interpretations The collision model of energy transfer The various “exact solutions” for atoms and molecules that BLP research claimed to have found The inner functionings of the orbitsphere model and the notion of the trap photon. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What I Won’t Talk About Experimental evidence and their interpretations The collision model of energy transfer The various “exact solutions” for atoms and molecules that BLP research claimed to have found The inner functionings of the orbitsphere model and the notion of the trap photon. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What I Will Present A critical analysis of the foundations of CQM with regards to: inner consistency of the theory the mathematical derivation of the electron model the mathematical intuition behind the theory Unless otherwise stated, material below pertaining to CQM is all taken from Randell Mills’ book, The Grand Uniﬁed Theory of Classical Quantum Mechanics, available free for download on BlackLight Power’s website. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What I Will Present A critical analysis of the foundations of CQM with regards to: inner consistency of the theory the mathematical derivation of the electron model the mathematical intuition behind the theory Unless otherwise stated, material below pertaining to CQM is all taken from Randell Mills’ book, The Grand Uniﬁed Theory of Classical Quantum Mechanics, available free for download on BlackLight Power’s website. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What I Will Present A critical analysis of the foundations of CQM with regards to: inner consistency of the theory the mathematical derivation of the electron model the mathematical intuition behind the theory Unless otherwise stated, material below pertaining to CQM is all taken from Randell Mills’ book, The Grand Uniﬁed Theory of Classical Quantum Mechanics, available free for download on BlackLight Power’s website. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM A bit of history The Theory: Does it Work? Overview of the claims Some other objections Overview Summary What I Will Present A critical analysis of the foundations of CQM with regards to: inner consistency of the theory the mathematical derivation of the electron model the mathematical intuition behind the theory Unless otherwise stated, material below pertaining to CQM is all taken from Randell Mills’ book, The Grand Uniﬁed Theory of Classical Quantum Mechanics, available free for download on BlackLight Power’s website. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Equation of Motion?! Some sophisticated mumble-jumble and jargon throwing: From The Grand Uniﬁed Theory of CQM, Chapter 1 One-electron atoms include the hydrogen atom. . . The mass- energy and angular momentum of the electron are constant; this requires that the equation of motion of the electron be tempo- rally and spatially harmonic. Thus, the classical wave equation applies and 2 1 ∂2 − 2 2 ρ(r , θ, φ, t) = 0 (1) v ∂t where ρ(r , θ, φ, t) is the function of the electron in time and space . . . All forces are central and Special Relativity applies. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Equation of Motion?! He includes two helpful footnotes: Footnote 1 The equation of motion of an extended electron is postulated based on ﬁrst principles and should not be confused with the en- ergy equation of a point-particle-probability-density wave such as the Schrödinger equation of quantum mechanics. and another about the classical wave equation Footnote 2 This is not to be confused with the Schrödinger equation which is not a proper wave equation. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Equation of Motion?! Two speciﬁc objections Contrary to the claims, there is no Lorentz invariance of the equation of motion. The phase velocity v in his equation would undergo a classical Lorentz transform in a new coordinate system: v → (v − w)c 2 /(c 2 − wv ) Hence the equation is only invariant if v = c. Furthermore, just conservation of energy and angular momenta doesn’t guarantee a classical wave equation: The trajectory of Newton’s ﬁrst law, x = x0 + vt perfectly conserves both. So from the very start, the theoretical underpinning of this theory is suspect. But we’ll take the leap of faith here and suppose the equation of motion a priori. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Equation of Motion?! Two speciﬁc objections Contrary to the claims, there is no Lorentz invariance of the equation of motion. The phase velocity v in his equation would undergo a classical Lorentz transform in a new coordinate system: v → (v − w)c 2 /(c 2 − wv ) Hence the equation is only invariant if v = c. Furthermore, just conservation of energy and angular momenta doesn’t guarantee a classical wave equation: The trajectory of Newton’s ﬁrst law, x = x0 + vt perfectly conserves both. So from the very start, the theoretical underpinning of this theory is suspect. But we’ll take the leap of faith here and suppose the equation of motion a priori. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Equation of Motion?! Two speciﬁc objections Contrary to the claims, there is no Lorentz invariance of the equation of motion. The phase velocity v in his equation would undergo a classical Lorentz transform in a new coordinate system: v → (v − w)c 2 /(c 2 − wv ) Hence the equation is only invariant if v = c. Furthermore, just conservation of energy and angular momenta doesn’t guarantee a classical wave equation: The trajectory of Newton’s ﬁrst law, x = x0 + vt perfectly conserves both. So from the very start, the theoretical underpinning of this theory is suspect. But we’ll take the leap of faith here and suppose the equation of motion a priori. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Equation of Motion?! Two speciﬁc objections Contrary to the claims, there is no Lorentz invariance of the equation of motion. The phase velocity v in his equation would undergo a classical Lorentz transform in a new coordinate system: v → (v − w)c 2 /(c 2 − wv ) Hence the equation is only invariant if v = c. Furthermore, just conservation of energy and angular momenta doesn’t guarantee a classical wave equation: The trajectory of Newton’s ﬁrst law, x = x0 + vt perfectly conserves both. So from the very start, the theoretical underpinning of this theory is suspect. But we’ll take the leap of faith here and suppose the equation of motion a priori. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The δ Function Ansatz From The Grand Uniﬁed Theory of CQM, Chapter 1 In general, radial solutions of the Helmholtz wave equation are spherical Bessel functions, Neumann functions, Hankel func- tions, and associated Laguerre functions. It was found that any radial solution with radial motion results in radiation. Thus, a solution of the two-dimensional wave equation plus time is the proper stable, nonradiative equation of motion of the bound elec- tron. The corresponding radial function is the radial Dirac delta function. Which basically is an assertion that a solution for the wave equation exists in the form 1 ρ(r , θ, φ, t) = 2 δ(r − rn )Ψ(θ, φ, t) (2) r Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The δ Function Ansatz: Some problems There are basically three problems with Mills’ reasoning: The various functions he cited are solutions to the Helmholtz differential equation 2 φ + k 2φ = 0 and not the wave equation in general. A charge density with no radial motion does not imply a chanrge density with the radial part represented by a delta function. The assertion that the δ-function ansatz gives rise to a solution to the wave equation is false. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The δ Function Ansatz: Some problems There are basically three problems with Mills’ reasoning: The various functions he cited are solutions to the Helmholtz differential equation 2 φ + k 2φ = 0 and not the wave equation in general. A charge density with no radial motion does not imply a chanrge density with the radial part represented by a delta function. The assertion that the δ-function ansatz gives rise to a solution to the wave equation is false. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The δ Function Ansatz: Some problems There are basically three problems with Mills’ reasoning: The various functions he cited are solutions to the Helmholtz differential equation 2 φ + k 2φ = 0 and not the wave equation in general. A charge density with no radial motion does not imply a chanrge density with the radial part represented by a delta function. The assertion that the δ-function ansatz gives rise to a solution to the wave equation is false. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The δ Function Ansatz: not actually a solution ρ(r , θ, φ, t) = (−v 2 ∂t2 + )ρ(r , θ, φ, t) (3) 2 1 = (−v 2 ∂t2 + ∂r2 + ∂r + 2 θ,φ )ρ(r , θ, φ, t)(4) r r 2 2 1 = Ψ(θ, φ, t)(∂r + ∂r ) 2 δrn (r ) (5) r r 1 1 + 2 δrn (r )(−v 2 ∂t2 + 2 θ,φ )Ψ(θ, φ, t) (6) r r Next we use a well-known theorem in mathematics, that the various derivatives of the Dirac δ function are linearly independent, with which we can separate the above expression into three terms. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The δ Function Ansatz: not actually a solution 1 ∂2 Ψ(θ, φ, t) δr (r ) = 0 (7) r 2 ∂r 2 n 2 ∂ Ψ(θ, φ, t) 3 δrn (r ) = 0 (8) r ∂r 2 1 1 Ψ(θ, φ, t) 4 + 2 (−v 2 ∂t2 + 2 θ,φ )Ψ(θ, φ, t) = 0 (9) r r rn The ﬁrst two of the above implies that Ψ = 0 throughout–quite different from what Mills obtained. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Haus Condition: What is it? In 1985, H.A. Haus: alternate interpretation of radiation. Haus was unaware of earlier work by Goedecke, Sommerﬁeld, Bohm, Weinstein, and more. The interpretation: radiation of charge/current distribution implies it has Fourier components synchronous with light-speed waves. The derivation is a simple exercise in taking Fourier transforms. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Haus Condition: What is it? In 1985, H.A. Haus: alternate interpretation of radiation. Haus was unaware of earlier work by Goedecke, Sommerﬁeld, Bohm, Weinstein, and more. The interpretation: radiation of charge/current distribution implies it has Fourier components synchronous with light-speed waves. The derivation is a simple exercise in taking Fourier transforms. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Haus Condition: What is it? In 1985, H.A. Haus: alternate interpretation of radiation. Haus was unaware of earlier work by Goedecke, Sommerﬁeld, Bohm, Weinstein, and more. The interpretation: radiation of charge/current distribution implies it has Fourier components synchronous with light-speed waves. The derivation is a simple exercise in taking Fourier transforms. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Haus Condition: What is it? In 1985, H.A. Haus: alternate interpretation of radiation. Haus was unaware of earlier work by Goedecke, Sommerﬁeld, Bohm, Weinstein, and more. The interpretation: radiation of charge/current distribution implies it has Fourier components synchronous with light-speed waves. The derivation is a simple exercise in taking Fourier transforms. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Haus Condition: What is it? Theorem (Goedecke 1964) Let J be the Fourier transform of a charge-current distribution (ρ, j). A sufﬁcient condition for nonradiation is that 2πn J( ˆ ˆ x , n) ∝ x , n>0 T where T is the period of the motion of the distribution (the ˆ distribution is localised in space), n an integer, x a unit vector. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Haus Condition: But is it enough? Pearle ’77 has an improved necessary and sufﬁcient result that uses the condition as k 0 J a (k ) = k a J 0 (k ) (10) where k is null. Mills’ application of Goedecke’s result is consistent with Pearle’s result, so modulo a constant factor, gives his claimed “boundary condition”. Pearle’s condition is not consistent with Special relativity. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Haus Condition: But is it enough? Pearle ’77 has an improved necessary and sufﬁcient result that uses the condition as k 0 J a (k ) = k a J 0 (k ) (10) where k is null. Mills’ application of Goedecke’s result is consistent with Pearle’s result, so modulo a constant factor, gives his claimed “boundary condition”. Pearle’s condition is not consistent with Special relativity. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM Wave equation The Theory: Does it Work? δ function Some other objections Haus condition Summary The Haus Condition: But is it enough? Pearle ’77 has an improved necessary and sufﬁcient result that uses the condition as k 0 J a (k ) = k a J 0 (k ) (10) where k is null. Mills’ application of Goedecke’s result is consistent with Pearle’s result, so modulo a constant factor, gives his claimed “boundary condition”. Pearle’s condition is not consistent with Special relativity. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Other objections Agreement with experiment? Closed form? Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Other objections Agreement with experiment? Closed form? Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Summary The theoretical foundation of CQM is shaky at best. The egregious “derivations” in the introductory chapters indicates that care must be taken to conﬁrm the calculations in later chapters. Unclear whether Mills’ experiments have revealed some yet-unknown property of our universe. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Summary The theoretical foundation of CQM is shaky at best. The egregious “derivations” in the introductory chapters indicates that care must be taken to conﬁrm the calculations in later chapters. Unclear whether Mills’ experiments have revealed some yet-unknown property of our universe. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Summary The theoretical foundation of CQM is shaky at best. The egregious “derivations” in the introductory chapters indicates that care must be taken to conﬁrm the calculations in later chapters. Unclear whether Mills’ experiments have revealed some yet-unknown property of our universe. Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Further References and Resources http://www.blacklightpower.com/ (where the GUT-CQM is free for download) GH Goedecke, “Classically Radiationless Motions and Possible Implications for Quantum Theory”, Phys. Rev. 135 (1B) July 1964, pB281 HA Haus, “On the radiation from point charges”, Am. J. Phys. 54 (12), December 1986, p1126 P Pearle, “Classical electron models” in Electromagnetism, Paths to Research, Plenum Press, NY, 1982; p211 A Rathke, “A critical analysis of the hydrino model”, New J. Phys. 7 (2005) 127 Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Further References and Resources http://www.blacklightpower.com/ (where the GUT-CQM is free for download) GH Goedecke, “Classically Radiationless Motions and Possible Implications for Quantum Theory”, Phys. Rev. 135 (1B) July 1964, pB281 HA Haus, “On the radiation from point charges”, Am. J. Phys. 54 (12), December 1986, p1126 P Pearle, “Classical electron models” in Electromagnetism, Paths to Research, Plenum Press, NY, 1982; p211 A Rathke, “A critical analysis of the hydrino model”, New J. Phys. 7 (2005) 127 Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Further References and Resources http://www.blacklightpower.com/ (where the GUT-CQM is free for download) GH Goedecke, “Classically Radiationless Motions and Possible Implications for Quantum Theory”, Phys. Rev. 135 (1B) July 1964, pB281 HA Haus, “On the radiation from point charges”, Am. J. Phys. 54 (12), December 1986, p1126 P Pearle, “Classical electron models” in Electromagnetism, Paths to Research, Plenum Press, NY, 1982; p211 A Rathke, “A critical analysis of the hydrino model”, New J. Phys. 7 (2005) 127 Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Further References and Resources http://www.blacklightpower.com/ (where the GUT-CQM is free for download) GH Goedecke, “Classically Radiationless Motions and Possible Implications for Quantum Theory”, Phys. Rev. 135 (1B) July 1964, pB281 HA Haus, “On the radiation from point charges”, Am. J. Phys. 54 (12), December 1986, p1126 P Pearle, “Classical electron models” in Electromagnetism, Paths to Research, Plenum Press, NY, 1982; p211 A Rathke, “A critical analysis of the hydrino model”, New J. Phys. 7 (2005) 127 Willie W. Wong Mathematical Evaluation of CQM Introduction: What is CQM The Theory: Does it Work? Some other objections Summary Further References and Resources http://www.blacklightpower.com/ (where the GUT-CQM is free for download) GH Goedecke, “Classically Radiationless Motions and Possible Implications for Quantum Theory”, Phys. Rev. 135 (1B) July 1964, pB281 HA Haus, “On the radiation from point charges”, Am. J. Phys. 54 (12), December 1986, p1126 P Pearle, “Classical electron models” in Electromagnetism, Paths to Research, Plenum Press, NY, 1982; p211 A Rathke, “A critical analysis of the hydrino model”, New J. Phys. 7 (2005) 127 Willie W. Wong Mathematical Evaluation of CQM