A quaternionic mass defining longitudinal mass () and transverse mass () of a particle is introduced. The longitudinal and transverse masses represent the mass of the particle and matter wave accompanying it, respectively. A massive particle is found to have a bigger longitudinal mass than transverse mass. The particle nature due to the transverse mass of a particle is described by a scalar wave satisfying the Klein-Gordon equation. The transverse mass vector is found to be along the direction of the particle’s velocity (). A particle of an equal longitudinal and transverse mass travels at speed of light in vacuum. The transmission of matter wave energy is found to be opposite to the particle velocity. The momentum of the particle () and its matter wave () are equal. The direction of the transverse mass is found to be along the direction of the particle’s velocity.
A quaternionic mass defining longitudinal mass () and transverse mass () of a particle is introduced. The longitudinal and transverse masses represent the mass of the particle and matter wave accompanying it, respectively. A massive particle is found to have a bigger longitudinal mass than transverse mass. The particle nature due to the transverse mass of a particle is described by a scalar wave satisfying the Klein-Gordon equation. The transverse mass vector is found to be along the direction of the particle’s velocity (). A particle of an equal longitudinal and transverse mass travels at speed of light in vacuum. The transmission of matter wave energy is found to be opposite to the particle velocity. The momentum of the particle () and its matter wave () are equal. The direction of the transverse mass is found to be along the direction of the particle’s velocity.
In the wave-particle duality, a free particle can be considered as a wave packet. Is there a wave property that corresponds to the rest mass of the particle? This is an interesting question that has not been extensively explored before. We suggest that this problem may be approached by treating the rest mass on the same footing as energy and momentum. Here we demonstrate that, by assuming that the matter wave of a particle behaves like a photon, one could derive the mass-energy relation from the solution of a simplified wave equation. This solution suggests that the rest mass of a particle is associated with a "transverse wave number", which characterizes the radial variation of the wave function in the transverse plane. This finding has several appealing features. For example, it shows a consistent geometrical relationship between mass, energy and momentum in both the wave and particle perspectives. Also, it predicts that a massless particle must travel at the constant speed of light. Furthermore, from a philosophical point of view, this model suggests that the mass-energy relation attributed to the special theory of relativity may have a root in the wave-particle duality. Indeed, it is shown that the Klein-Gordon equation can be derived naturally based on this model.
In this paper, we have considered that the particle velocity in the matter wave concept of de Broglie is relative. It reveals that matter wave associated with a moving particle is a photon wave which has longitudinal and transverse values. The conventional de-Broglie’s wavelength-momentum relation corresponds to the longitudinal wave. As a consequence, we have found the longitudinal and transverse relative values of the mass, time, length and velocity including the missing equations of Einstein’s theory of special relativity. It also reveals that momentum and energy are relative quantities and the corresponding expressions for the longitudinal and transverse values for both of them have been presented. Moreover, the addition rules, with a general expression, for all the relative quantities based on Lorentz-Einstein transformation law for the addition of velocities have been given. Finally, the maximum and minimum values of the relative quantities have been expressed in terms of Plank’s constant and the speed of light. Their numerical values have also been calculated and presented in this paper.
Abstract Various attempts at a thoroughly wave-theoretic explanation of matter have taken as their fundamental ingredient the de Broglie or matter wave. But that wave is superluminal, whereas it is implicit in the Lorentz transformation that influences propagate ultimately at the velocity c of light. It is shown that if the de Broglie wave is understood, not as a wave in its own right, but as the relativistically induced modulation of an underlying standing wave comprising counter-propagating ináuences of velocity c, the energy, momentum, mass and inertia of a massive particle can be explained from the manner in which the modulated wave structure must adapt to a change of inertial frame. With those properties of the particle explained entirely from wave structure, nothing remains to be apportioned to anything discrete or ìsolidîwithin the wave. Consideration may thus be given to the possibility of wave-theoretic explanations of particle trajectories, and to a deeper understanding of the Klein-Gordon, Schrˆdinger and Dirac equations, all of which were conceived as equations for the de Broglie wave.
Momentum and Kinetic Energy equations are developed from the hypothesis that oppositely directed components of harmonically oscillating pseudo standing waves pass through a quantum particle center and can be represented by Longitudinal Matter Waves that carry the particle’s momentum and energy. The Doppler effect on the component wave lengths allows the net forward momentum and kinetic energy to increase with speed well beyond classical values. De Broglie (1925) issues with stationary wavelength and moving pulse rate are resolved in a different manner. Because a quantum particle is considered to be nothing more than the sum of “in” and “out” matter waves focused through its center, whatever happens to these matter waves determines the future location of that center. This opens the door to physical explanations for gravity, interference, and the slowdown of light in transparent mediums. Gravity, for example, is shown in section 6, to possibly be caused by the local gradient in matter wave speed near a large body like earth.
The velocity is a macroscopic concept. The question what is the meaning of velocity in Quantum Mechanics( QM) ,this problem itself is inno answersituation. In the semi-classical theory,for the reconstruction of the velocity definition,we have v→= ▽S / m[m is the particle's mass and ψ = Aexp( j S /h) ]. This method builds the relation of velocity and wave function,but it is not the necessary and useful concept of wave's speed. The velocity of light in vacuum is a common basic constant of physics,however QM also need this parameter obviously,i. e. the new wave mechanics is on need of this physical quantity.The process and result of people's practice show that,scientist has intended to bring forth in the measurement of c with very high accuracy does stand out,if he used the scalar equation c = fλ. Modern physics maintain that the wave velocities( phase velocityvp、group velocity vg) are scalars,not vectors.The wave has its special scientific features; the regular pattern is different from the Classical Mechanics( CM). It needs supports of the QM,but it is not identical with the serious QM. Now,it has not only a theory that can explain the wave problems perfectly,so it is support by the classical physics and quantum physics.The quantization process of EM fields explains the particle-wave duality mathematically. Post quantization,from the EM fields we obtained the photons. But scientist still is puzzled by a difficult question:what is photon. As the common knowledge,the matter waves of photon still a paradox in physics. In addition,the electronic probability waves corresponding the Schrdinger equation,but people says the photonic probability waves have not that wave equation. In other words,the physical meanings of matter wave's phase velocity are not clear,and why the phase velocity of de Broglie waves is faster-than-light,many explanations will be needed.Many instances similar to these examples make the particle-wave duality still the magic ghost in modern physics. Finally,the study on negative wave velocity( NWV) in recent years has a great deal of experience on wave sciences,and bring the new chance on theoretical thinking.