PUH Theorem 187 — The de Broglie Wavelength from E8 Lattice Bow Wave Mechanics: lambda = h/p as the Correlation Length of a Vortex Moving Through the E8 Phase I Condensate, and the Mechanical Resolution of Wave-Particle Duality
In 1924 Louis de Broglie proposed that every particle of momentum p has an associated wavelength lambda = h/p, by analogy with Einstein’s E = hν for photons. The proposal was confirmed experimentally by Davisson and Germer (1927) via electron diffraction. But de Broglie could not explain the mechanism: why does a massive particle have a wavelength? What is the wave? What is waving? For 100 years these questions have had no mechanical answer. This paper establishes that within the Photonic Universe Hypothesis, lambda = h/p is the correlation length of an E8 lattice pressure wave (a bow wave) generated by a localised vortex pattern moving through the E8 Phase I condensate. The mechanism is exact: (1) A particle is a localised vortex in the E8 Phase I lattice. (2) As the vortex moves at velocity v, it displaces lattice nodes ahead of it, generating a pressure wave that propagates at the E8 phase-mode wave speed c. (3) The wavelength of this bow wave is lambda = h/p = h/(mv), where h is the minimum E8 lattice displacement (Theorem 176) and p = mv is the momentum of the vortex pattern. Wave-particle duality is resolved without mystery: the particle is the vortex (localised, follows a definite trajectory, detected at a point); the wave is the E8 lattice pressure wave accompanying the vortex (delocalised, produces interference, passes through both slits). They are not the same object. They are two aspects of one E8 phenomenon: a localised vortex moving through a continuous 248-dimensional condensate. The double-slit experiment is explained mechanically. Davisson-Germer electron diffraction is the E8 lattice bow wave diffracting off a crystal lattice. Wave function collapse is vortex absorption — no probability collapse occurs. The Bohmian pilot wave is the E8 bow wave; the quantum potential is the E8 lattice pressure gradient. Four falsifiable predictions follow.
In 1924 Louis de Broglie proposed that every particle of momentum p has an associated wavelength lambda = h/p, by analogy with Einstein’s E = hν for photons. The proposal was confirmed experimentally by Davisson and Germer (1927) via electron diffraction. But de Broglie could not explain the mechanism: why does a massive particle have a wavelength? What is the wave? What is waving? For 100 years these questions have had no mechanical answer. This paper establishes that within the Photonic Universe Hypothesis, lambda = h/p is the correlation length of an E8 lattice pressure wave (a bow wave) generated by a localised vortex pattern moving through the E8 Phase I condensate. The mechanism is exact: (1) A particle is a localised vortex in the E8 Phase I lattice. (2) As the vortex moves at velocity v, it displaces lattice nodes ahead of it, generating a pressure wave that propagates at the E8 phase-mode wave speed c. (3) The wavelength of this bow wave is lambda = h/p = h/(mv), where h is the minimum E8 lattice displacement (Theorem 176) and p = mv is the momentum of the vortex pattern. Wave-particle duality is resolved without mystery: the particle is the vortex (localised, follows a definite trajectory, detected at a point); the wave is the E8 lattice pressure wave accompanying the vortex (delocalised, produces interference, passes through both slits). They are not the same object. They are two aspects of one E8 phenomenon: a localised vortex moving through a continuous 248-dimensional condensate. The double-slit experiment is explained mechanically. Davisson-Germer electron diffraction is the E8 lattice bow wave diffracting off a crystal lattice. Wave function collapse is vortex absorption — no probability collapse occurs. The Bohmian pilot wave is the E8 bow wave; the quantum potential is the E8 lattice pressure gradient. Four falsifiable predictions follow.
De Broglie-Bohm pilot wave theory (1927/1952) is a realist, deterministic formulation of quantum mechanics in which particles have definite positions at all times, guided by a pilot wave evolving according to the Schrodinger equation. It reproduces all predictions of standard quantum mechanics while restoring objective reality. Despite this, it has been sidelined for 74 years by one unanswerable question: what physical medium carries the pilot wave? Bohm was forced to place the wave in abstract configuration space, making the theory philosophically unsatisfying. This paper establishes that the E8 Phase I condensate is the physical substrate of the pilot wave. The particle is an E8 vortex (Theorem 187). The pilot wave is the E8 bow wave generated by that vortex moving through the Phase I condensate (Theorem 187). The guiding equation dQ/dt = (hbar/m)Im(psi nabla psi / psi* psi) is derived from E8 condensate hydrodynamics: writing psi = |psi|exp(iphi), the equation becomes dQ/dt = (hbar/m)nabla phi — the standard superfluid vortex velocity law. The E8 vortex moves at the local E8 condensate phase gradient, exactly as a quantised vortex in a BEC moves at the local superfluid velocity. The Bohm quantum potential Q_B = -(hbar^2/2m)(nabla^2|psi|)/|psi| is the E8 bow wave pressure curvature — the force on the vortex from the curved bow wave amplitude envelope, exactly as a surfer on curved ocean swell experiences physical pressure. The pilot wave resides in the E8 Phase I condensate — a real 248-dimensional physical space — not abstract configuration space. For N particles, N bow waves interact through the shared condensate; this is the physical origin of quantum non-locality. Entanglement is phase-locking of bow waves through the shared E8 condensate medium. The double-slit experiment is fully mechanical: bow wave through both slits, interference in the condensate, vortex guided to intensity maxima. PUH supplies what de Broglie and Bohm were missing: the ocean beneath the surfer. Six formal theorems and four falsifiable predictions follow.
In 1926 Erwin Schrodinger wrote down ihbar(d psi/dt) = [-(hbar^2/2m)nabla^2 + V(x)]psi by analogy with classical wave optics. He had no derivation. Three features have remained unexplained for 100 years: why first-order in time but second-order in space; why the imaginary unit i; and what psi physically represents. This paper derives the Schrodinger equation from the Photonic Universe Hypothesis without postulates. The wave function psi(x,t) is the complex amplitude of the E8 bow wave generated by a particle vortex moving through the E8 Phase I condensate (Theorem 187). The free Schrodinger equation follows from the E8 bow wave dispersion relation omega = hbark^2/(2m), derived from E = p^2/(2m) and the de Broglie relations E = hbaromega, p = hbark. The first-order/second-order asymmetry reflects quadratic (not linear) dispersion — a consequence of the non-relativistic energy-momentum relation for a massive vortex. The potential term V(x)psi arises from local E8 Phase I lattice tension variations lambda(x) that modify the bow wave phase velocity. The imaginary unit i is the U(1) phase rotation generator: quantum mechanics is complex because the E8 bow wave is a U(1) Cartan phase-mode oscillation (Theorem 167), and U(1) rotations are naturally represented by complex exponentials. The time-independent equation Hphi = Ephi gives standing E8 bow wave resonance patterns. The hydrogen energy levels E_n = -13.6 eV/n^2 are the resonant frequencies of the electron bow wave standing around the proton vortex. Chemistry is E8 bow wave resonance. Six formal theorems, the complete quantum mechanics audit table, and four falsifiable predictions follow.
The de Broglie wavelength λ= h/mv of simple particles (photon, electron, neutron) atoms (helium, neon) and molecules (fullerene, fluorous porphyrin) deduced from the de Broglie (dB) relation and those measured by double slits interferometry are compared.The de Broglie relation is verified in very large domains of velocity v and mass m of the particles, respectively 8 and 13 orders of magnitude.In this paper one analyses the interference spectra of photons and electrons and give the Intensity Relations IR(II-DI) between the interference intensity (II) by two slits and the diffracted intensity (DI) by one slit.The double slits interference in the de Broglie-Bohm interpretation is an interference of particle and pilot waves, internal and external waves each passing by a different slit.The velocities of these waves are discussed.From the universal de Broglie relation and the relation vp = vw between the particle and pilot waves one shows that only in the domain of the maximum of the light spectrum (λm~ 550 µm) the dB relation leads to the Plank-Einstein (PE) relation hν=mc 2 .A measure of the mass of the photon m* = 4 10 -36 Kg is deduced.The de Broglie relation for different wave lengths shows that the PE relation must be modified, the velocity of light is not constant as shown by various experiments: Global Positioning Systems GPS, red shift in astronomy etc.This has been discussed by Marmet in years 2000.The same conclusion was given by Sato in 2010 for electromagnetic waves.The trajectories of electrons observed by Jönsson by double slits interferometry and then calculated by Gondran according to the de Broglie-Bohm interpretation are then analyzed in detail.The correlations observed between the properties of these trajectories at short and long distances from the slits permit to give the general equation of the trajectories.The origin of these properties is discussed in the framework model of De Broglie and Bohm, assuming that at any point the intensity of the two waves (particle and pilot wave) passing by the 2 slits are deceasing functions of the distance to the slits.The changes of direction of the trajectories are explained.
This chapter describes the dual nature of electron/de Broglie waves. de Broglie suggested that like light, an electron also behaves as both a particle and a wave. Based on this hypothesis, a relationship between wavelength of electrons (a particle) and momentum was developed. It opines that ‘wavelength of the particle is inversely proportional to momentum’. This is called the de Broglie relation. On the basis of the above facts, we have developed a relationship between wavelength and voltage applied and this relation may be stated as ‘the wavelength of a particle is inversely proportional to the square root of the voltage applied to the system’. This chapter also uses the Davisson and Germer experiment to show the wave nature of the particle, i.e., diffraction of electrons by crystal lattice. The experimental results were the same as found in the case of the diffraction of X-ray by crystals. This chapter also provides an idea on quantisation of angular momentum. It also discusses Heisenberg’s uncertainty principle, phase velocity, group velocity, and uncertainty relation between energy and time. The chapter comprises the experimental evidence of Heisenberg’s uncertainty principle, such as by diffraction of electrons through a slit and gamma ray microscope thought experiment, which is shows the physical significance of the uncertainty principle. Sufficient references, solved problems, and questions on concepts have been provided at the end of chapter.
In 1927 Werner Heisenberg derived the uncertainty principle delta_x * delta_p >= hbar/2 from the commutation relation [x, p] = ihbar and the Cauchy-Schwarz inequality. He motivated it through the gamma-ray microscope thought experiment: measuring position precisely requires high-energy photons that impart momentum kicks. But the principle holds even for ideal measurements that cause no disturbance — it is a property of quantum states, not of measurement apparatus. The true origin is deeper: the commutation relation [x, p] = ihbar itself. What is its physical origin? Why ihbar and not zero? For 100 years this has had no mechanical answer. This paper establishes that within the Photonic Universe Hypothesis, the Heisenberg uncertainty principle is a geometric property of the E8 Phase I condensate. Position x and momentum p are canonically conjugate variables in the E8 Lie-Poisson bracket. The commutator [x, p] = ihbar is the quantisation of the classical Poisson bracket {x, p} = 1, with hbar = E_Pt_P being the minimum E8 lattice action (Theorem 176). The uncertainty product delta_x * delta_p >= hbar/2 is derived three independent ways: (1) from the E8 Lie-Poisson bracket and canonical quantisation; (2) from the Fourier duality of E8 bow waves (Theorem 187): sharp position requires a localised bow wave packet and sharp momentum requires a plane wave — these are mutually exclusive; (3) from the U(1) phase-number uncertainty of the E8 phase mode. The factor hbar/2 (not hbar) comes from the QHO ground state energy hbaromega/2, arising from the canonical commutation relation [a, a+] = 1 (Theorem 174). The energy-time uncertainty delta_E * delta_t >= hbar/2 follows from the E8 phase-mode dispersion relation E = hbar*omega. The uncertainty principle is not about measurement disturbance. It is about E8 lattice geometry. Four falsifiable predictions follow.
Bohr's response in 1925 to the failure of the Bohr–Kramers–Slater theory was, as we have seen, to regard wave–particle duality as having a formal, rather than realistic, significance; he saw it as a ‘limitation of our usual means of visualization’. That same year, however, wave–particle duality was generalised by Louis-Victor de Broglie to cover the theory of matter. Moreover, the advent of quantum mechanics (i.e. matrix mechanics and wave mechanics), rather than dispelling the duality problem, heightened it. How did Bohr respond to these new developments in the quantum theory? A matter of waves In the autumn of 1923 de Broglie put forward the extremely bold hypothesis that any particle whatever is associated with a wave whose wavelength λ and frequency ν are proportional to the particle's momentum and energy respectively: λ = h / p , v = E / h . De Broglie regarded his hypothesis as the basis of a new mechanics, a wave mechanics related to classical mechanics in the same way as wave optics is related to ray optics. On this view the trajectory of a particle corresponds to the ray of the wave associated with it (i.e. the normal to the equiphase surface of the wave), and the velocity of the particle corresponds to the group velocity of the wave. De Broglie suggested, moreover, that in certain circumstances particles should exhibit diffraction effects: a stream of electrons, for example, which passes through a slit in a screen should undergo diffraction.