It is well known that the kinetic energy formula of classical mechanics has a limited range of applicability. That formula breaks down when the mass of a moving body increases and the effects of relativity can no longer be ignored. More than a hundred years ago, Sommerfeld and Einstein defined relativistic kinetic energy as the difference between a body’s relativistic energy and its rest mass energy. This paper, however, derives a formula for relativistic kinetic energy that is not a definitional formula. The formula derived in this paper incorporates the phase velocity of a wave. It further finds that the phase velocity of the wave is an important physical quantity linking a body’s kinetic energy and momentum. In current standard physics, the de Broglie wave is not regarded as having physical reality. Nevertheless, this paper concludes that the phase velocity of a matter wave is an essential physical quantity.
It is well known that the kinetic energy formula of classical mechanics has a limited range of applicability. That formula breaks down when the mass of a moving body increases and the effects of relativity can no longer be ignored. More than a hundred years ago, Sommerfeld and Einstein defined relativistic kinetic energy as the difference between a body’s relativistic energy and its rest mass energy. This paper, however, derives a formula for relativistic kinetic energy that is not a definitional formula. The formula derived in this paper incorporates the phase velocity of a wave. It further finds that the phase velocity of the wave is an important physical quantity linking a body’s kinetic energy and momentum. In current standard physics, the de Broglie wave is not regarded as having physical reality. Nevertheless, this paper concludes that the phase velocity of a matter wave is an essential physical quantity.
Among the exciting outcomes of the Beaumanner relative quantum formula RQF is the linking of de Broglie's matter wave length to the velocity of light. I show that a phase velocity introduced by de Broglie is not believable since it comes from relating his matter wave length to the total Einstein energy. Instead it should relate to the energy of mechanical momentum which I have defined. In an additional argument I show that the strange phase velocity also can be viewed as coming from an assumption that mechanical momentum is linearly coupled to the total energy, which it is not.
Abstract This work revisits the apparent paradox that emerges from some results found in many basic textbooks which address the problem of calculating the phase velocity of free particles in the relativistic regime and its relation with the same quantity in the non-relativistic regime. Because some books use the Newtonian kinetic energy and others use the total relativistic energy (including the rest energy), the non-relativistic limit of the relativistic phase velocity differs from the Newtonian phase velocity, which can be confusing for many students. This paper proposes expressing the phase velocity by first subtracting the rest energy from the relativistic energy before imposing the usual de Broglie relations. In addition to providing the corresponding Newtonian phase velocity, it also provides a relativistic Schrödinger equation which intuitively reduces to the non-relativistic Schrödinger equation in the appropriate regime. The problem of the equivalence between the relativistic Schrödinger equation and the Klein–Gordon equation is also addressed by using arguments which can be understood by undergraduate students.
We will here show that there is one more relativistic wave equation rooted in the relativistic energy Compton moment relation, which should not be confused with the standard relativistic energy momentum relation. The standard momentum is, from a quantum perspective, rooted in the de Broglie wave. The de Broglie wave is not mathematically defined for rest-mass particles and has strange properties such as converging to infinity when the particle is almost at rest. As mentioned in the previous paper, the de Broglie wave is likely just a mathematical derivative of the true matter wave, which we have good reasons to think is the Compton wave. A wave equation that satisfies the relativistic energy Compton momentum relation will, in addition automatically satisfy the standard relativistic energy momentum relation, so there is no conflict between these two relations. The new one related to Compton is just the deeper reality and help explain why it gives simpler and more elegant relativistic wave equations, which are likely easier to interpret in terms of the physical reality.
In an Einstein like gedanken experiment, the relativistic changes of energies of a photon and a rest mass particle caused by the same relative movement were compared and it was found, that the same relative movement was causing very different energetic changes on the photons and rest mass particles: photons have significantly more “capacity” for kinetic energy. To explain this phenomena, which could be named kinetic capacity effect, a physical model of a “folded and looped photon” was created and checked quantitatively through relativistic Doppler effect for kinetic and total energies, and for linear momentum. De Broglie’s material wave was identified as modulation wave of two interfering waves of the correlated half photons inside of the particle, resulting in known group and phase velocities. Photons themselves were not changed in physical description, so Quantum Mechanics will stay in use with them, and Special Relativity has now extended abilities for entering into particles, opening a way through discussed model to unify both theoretical worlds. For the moment, the two classical theories, waves and relativity, have confirmed exact the hypothesis of present model.
Matter waves were discovered in the early 20th century from their wavelength, predicted by DeBroglie, Planck's constant divided by the particle's momentum, that is, lmw = h/mv. But, the failure to obtain a reasonable theory for the matter wave frequency resulted somewhat in loss of further interest. It was expected that the frequency of the matter wave should correspond to the particle kinetic energy, that is, fmw = 1/2mv^2/h but the resulting velocity of the matter of the particle, v = fmw x lmw, is that the matter wave moves at one half the speed of the particle, obviously absurd as the particle and its wave must move together. If relativistic mass is used (as it should in any case) the problem remains, the same mass appearing in numerator and denominator and canceling. It is no help to hypothesize that the total energy, not just the kinetic energy, yields the matter wave. That attributes a matter wave to a particle at rest. It also gives the resulting velocity as c^2/v, the wave racing ahead of its particle. A reinterpretation of Einstein's derivation of relativistic kinetic energy (which produced his famous E = mc^2) leads to a valid matter wave frequency and a new understanding of particle kinetics and of the atom's stable orbits.
Matter waves were discovered in the early 20th century from their wavelength, predicted by DeBroglie, Planck’s constant divided by the particle’s momentum, that is, λmw = h/m.v. But, the failure to obtain a reasonable theory for the matter wave frequency resulted somewhat in loss of further interest. It was expected that the frequency of the matter wave should correspond to the particle kinetic energy, that is, fmw = 1/2 . m . v2/h but the resulting velocity of the matter of the particle, v = fmw . λmw , is that the matter wave moves at one half the speed of the particle, obviously absurd as the particle and its wave must move together. If relativistic mass is used (as it should in any case) the problem remains, the same mass appearing in numerator and denominator and canceling. It is no help to hypothesize that the total energy, not just the kinetic energy, yields the matter wave. That attributes a matter wave to a particle at rest. It also gives the resulting velocity as c2/v the wave racing ahead of its particle. A reinterpretation of Einstein’s derivation of relativistic kinetic energy (which produced his famous E = m.c2 ) leads to a valid matter wave frequency and a new understanding of particle kinetics and of the atom’s stable orbits.