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On the Mechanical Properties of Matter Waves

Nicholas K Whitlock, Stephen Mark Barnett, John Jeffers

Published Jan 1, 2004• 3 citations
OpenAlex

Abstract

PACS number: 03.75.-b Abstract. Electromagnetic waves and fluids have locally conserved mechanical properties associated with them and we may expect these to exist for matter waves. We present a semiclassical description of the continuity equations relating to these conserved properties of matter-waves and derive a general expression for their respective fluxes. It has long been known that electromagnetic waves have associated mechanical properties. Maxwell described the radiation pressure exerted by light [1] and Poynting identified the momentum carried by light [2] and that angular momentum was associated with circular polarisation [3]. The energy, momentum and angular momentum of electromagnetic waves are particularly important as these quantities are all conserved. Associated with the density of each of these quantities is a flux density (or current density) which is related to the density via continuity equations [4]. The creation of the first Bose-Einstein condensates (BECs) in dilute atomic gases in 1995 [5, 6, 7] brought about the possibility of creating coherent matter waves. A BEC in a trap is analogous to an electromagnetic field in a laser cavity, in that macroscopic occupation of a single quantum state occurs. Just as a laser cavity is a source of coherent light, a BEC can be a source of coherent matter waves. To continue this analogy we assert that matter waves will have associated mechanical properties and present a general theory of the local mechanical properties of matter waves. We give, as examples, the densities and flux densities of the matter wave particle number and energy. The same theory applies to their momentum and angular momentum. The local conservation of particles is a well established concept in fluid mechanics. It states that in any volume element of a fluid, no matter how small, the rate of change of particles in an element of fluid is equal to the rate at which the particles flow into the surface of the element. This principle is made quantitative in the following equation [8];

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On the mechanical properties of matter waves

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In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. 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On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. 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Archives of physics research•Jan 1, 2018•0 citations
PACS number: 03.75.-b Abstract. Electromagnetic waves and fluids have locally conserved mechanical properties associated with them and we may expect these to exist for matter waves. We present a semiclassical description of the continuity equations relating to these conserved properties of matter-waves and derive a general expression for their respective fluxes. It has long been known that electromagnetic waves have associated mechanical properties. Maxwell described the radiation pressure exerted by light [1] and Poynting identified the momentum carried by light [2] and that angular momentum was associated with circular polarisation [3]. The energy, momentum and angular momentum of electromagnetic waves are particularly important as these quantities are all conserved. Associated with the density of each of these quantities is a flux density (or current density) which is related to the density via continuity equations [4]. The creation of the first Bose-Einstein condensates (BECs) in dilute atomic gases in 1995 [5, 6, 7] brought about the possibility of creating coherent matter waves. A BEC in a trap is analogous to an electromagnetic field in a laser cavity, in that macroscopic occupation of a single quantum state occurs. Just as a laser cavity is a source of coherent light, a BEC can be a source of coherent matter waves. To continue this analogy we assert that matter waves will have associated mechanical properties and present a general theory of the local mechanical properties of matter waves. We give, as examples, the densities and flux densities of the matter wave particle number and energy. The same theory applies to their momentum and angular momentum. The local conservation of particles is a well established concept in fluid mechanics. It states that in any volume element of a fluid, no matter how small, the rate of change of particles in an element of fluid is equal to the rate at which the particles flow into the surface of the element. This principle is made quantitative in the following equation [8];
Further development of the method of quantum hydrodynamics in application for Bose-Einstein condensates (BECs) is presented. To consider evolution of polarization direction along with particles movement we have developed corresponding set of quantum hydrodynamic equations. It includes equations of the polarization evolution and the polarization current evolution along with the continuity equation and the Euler equation (the momentum balance equation). Dispersion properties of the transverse waves including the electromagnetic waves propagating through the BECs are considered. To this end we consider full set of the Maxwell equations for description of electromagnetic field dynamics. This approximation gives us possibility to consider the electromagnetic waves along with the matter waves. We find a splitting of the electromagnetic wave dispersion on two branches. As a result we have four solutions, two for the electromagnetic waves and two for the matter waves, the last two are the concentration-polarization waves appearing as a generalization of the Bogoliubov mode. We also obtain that if the matter wave propagate perpendicular to external electric field when dipolar contribution does not disappear (as it follows from our generalization of the Bogoliubov spectrum). In this case exist a small dipolar frequency shift due to transverse electric field of perturbation.
Why does matter possess wave-like properties? Is wave-particle duality an inherent dual nature of matter, or merely a projection of a deeper underlying physical structure under two distinct observational perspectives? This paper presents a rigorous first-principles derivation of matter wave dynamics within the axiomatic framework of Time Field Theory (TFT). The entire derivation relies solely on two core axioms: local intrinsic total velocity conservation and metric self-consistency (flux conservation), without introducing any extra postulates. Following the logical sequence of ontological foundation, phase derivation, wave generation, hierarchical classification, physical correspondence, extreme-case verification, and unified spectrum construction, this paper first establishes the core ontological premise that matter corresponds to localized condensed states of the time field. It then derives the geometric structure of the existence phase angle on the unit circle of the complex plane from the velocity conservation axiom, revealing the fundamental origin of intrinsic material phases. On this basis, the de Broglie relations are strictly derived, eliminating the dualistic contradiction between particle and wave characteristics. This work further distinguishes two distinct wave regimes: intrinsic waves originating from phase evolution within condensed states, and apparent waves arising from the global spatial translation of field distributions, which are respectively mapped onto electromagnetic waves and gravitational waves as two classes of physical reality. Using black hole horizons as an extreme physical testbed, we demonstrate that intrinsic wave phenomena vanish completely at time-freezing surfaces while apparent wave behaviors remain intact, providing a unified explanation for the observational fact that black holes emit no light yet produce gravitational waves. We also clarify the ontological origin of black hole darkness: light is not trapped by gravity, but the physical conditions required for photon generation cease to hold at time-freezing surfaces. Finally, we construct a continuous spectrum of wave behaviors ranging from the vacuum ground state to black hole horizons. This paper proves that wave-like behavior is not a binary ”present/absent” attribute of matter, but a continuous function of the condensation strength of the time field. Quantum mechanics and classical mechanics do not describe two separate physical worlds; instead, they represent two limiting regimes of a single universal time field under different condensation levels.
Abstract. An alternative description of quantum scattering processes rests on inhomogeneous terms amended to the Schrödinger equation. We detail the structure of sources that give rise to multipole scattering waves of definite angular momentum, and introduce pointlike multipole sources as their limiting case. Partial wave theory is recovered for freely propagating particles. We obtain novel results for ballistic scattering in an external uniform force field, where we provide analytical solutions for both the scattering waves and the integrated particle flux. As an illustration of the theory, we predict some properties of vortex-bearing atom laser beams outcoupled from a rotating Bose–Einstein condensate under the influence of gravity. PACS numbers. 03.65.Nk — Scattering theory 03.75.-b — Matter waves 03.75.Fi — Phase coherent atomic ensembles; quantum condensation phenomena
The continuity equation: d/dt partial d(x,t) + grad dot d(x,t)u(x,t) = 0 ((1)) where d(x,t) is density and u(x,t) a velocity field appears in the literature as a fundamental equation of “continuity”. For example, it follows from Boltzmann’s transport equation (and conservation of mass) and leads to a wave equation for sound when combined with a pressure equation (1). A continuity type equation also follows from Maxwell’s em equations and includes the Poynting momentum b El x B where b is a constant, El electric field and B magnetic field. This equation describes light. Finally the continuity equation is used in a number of derivations of the Schrodinger equation (e.g. (2)). In previous notes (3) we have argued that two flux/flow equations for free particles: d/dx partial A = p ((2a)) and d/dt partial A = -E ((2b)), with A=-Et+px being a solution, hold for both quantum particles with rest mass and without (e.g. photons/phonons). These equations are also involved with flux which begs the question: What is the relationship between the two? We investigate this in this note.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.