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Book Publisher International (a part of SCIENCEDOMAIN International)

Reconsideration of Matter Waves: A Brief Study

Roger Ellman

Published Aug 24, 2021• 0 citations
Publisher

Abstract

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, \(\lambda\)mw = 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 . \(\lambda\)mw , 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.

DOI
10.9734/bpi/nupsr/v15/3600f

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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, (\lambda)mw = 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 . (\lambda)mw , 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.
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.
Louis-Victor de Broglie secured his place in the history of quantum physics by proposing the existence of matter waves. Following the concept accepted early in the twentieth century that light waves could display properties characteristic of particles (photons), de Broglie suggested the complementary principle that particles could display properties characteristic of waves. Within a few years, this far-reaching proposal was experimentally verified. The well-known formula for the wavelength λ of the wave associated with a particle of momentum p, namely λ=h/p, does not appear in de Broglie’s original paper, which was published in this journal in 1924. What is derived from the starting assumption that the frequency of the wave is the rest energy of the particle divided by Planck’s constant h, is the wave's velocity, which turns out to be c2/v, where v is the particle’s velocity. This velocity, being superluminal, means that the wave cannot transport energy. De Broglie called it a phase wave and postulated a group of such waves having virtually, but not exactly, the same velocity and wavelength. The group velocity of such a large number of almost identical phase waves is then shown, very satisfactorily, to be equal to the velocity of the particle. The proposal by de Broglie inspired Schrödinger to develop the mathematical foundations of wave mechanics. Whereas in conventional quantum theory, phase waves are considered fictional, de Broglie considered them to have a certain physical reality, renaming them pilot or guiding waves – a concept developed later by Bohm in 1952.
The atom's orbital electron structure in terms of quantum numbers (principal, azimuthal, magnetic and spin) results in space for a maximum of: 2 electrons in the n=1 orbit, 8 electrons in the n=2 orbit, 18 electrons in the n=3 orbit, and so on. Those dispositions are correct, but that is not because of quantum numbers nor angular momentum nor a "Pauli exclusion principle". Matter waves were discovered in the early 20th century from their wavelength, which was predicted by DeBroglie to be, Planck's constant divided by the particle's momentum. But, the failure to obtain a reasonable theory for the matter wave frequency resulted in loss of interest. That problem is resolved in "A Reconsideration of Matter Waves" in which 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 the atom's stable orbits. It is analytically shown that the orbital electron arrangement is enforced by the necessity of accommodating the space that each orbiting electron's matter wave occupies.
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 analytically shown that the orbital electron arrangement is enforced by the necessity of accommodating the space that each orbiting electron's matter wave occupies. The atom's orbital electron structure in terms of quantum numbers (principal, azimuthal, magnetic and spin) results in space for a maximum of: 2 electrons in the n=1 orbit, 8 electrons in the n=2 orbit, 18 electrons in the n=3 orbit, and so on. Those dispositions are correct, but that is not because of quantum numbers nor angular momentum nor a "Pauli exclusion principle". Matter waves were discovered in the early 20th century from their wavelength, which was predicted by DeBroglie to be, Planck's constant divided by the particle's momentum, λ\lambdaλmw = h/m . v . But, the failure to obtain a reasonable theory for the matter wave frequency resulted in loss of interest. That problem is resolved in "A Reconsideration of Matter Waves" 2 in which 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 the atom's stable orbits. It is analytically shown that the orbital electron arrangement is enforced by the necessity of accommodating the space that each orbiting electron's matter wave occupies.
The long history of the dispute among physicists as to whether matter is composed of particles or waves is reviewed. A turning point came at the dawn of the 20th century in the form of the de Broglie momentum-wavelength and Bohr/Planck energy frequency relations. It was on this basis that Schrödinger developed a quantum mechanical differential equation whose solution came to be known as a wave function. The Born Interpretation of these functions holds that their absolute square constitutes a probability distribution which has been used successfully to make predictions of the properties of the system under discussion. It is pointed out that the phenomenon of light refraction played a key role in the development of both Newton’s corpuscular theory of light and the competing theory of Huygens which looks upon light as consisting exclusively of waves. It is shown that Newton’s failure to predict the decrease in the speed of light as it passes from air into water was not because of his belief in the particle composition of light, but rather because he not did anticipate that the mass of a photon changes upon entering a medium of higher index of refraction n. By assuming that the energy E of light is equal pc/n, where p is the momentum of the photons and c is the speed of light in free space, it is shown that the experimental dependence of the speed the light on its wavelength as it passes through a transparent medium is derived successfully through the use of Hamilton’s Canonical Equations and Newton’s Second Law of Kinetics. It is suggested that the relationship between particles and waves can be understood by noting that the localized properties of matter exhibit themselves in experiments such as the photoelectric effect where attention can be concentrated on the behavior of a single particle. The corresponding wave properties occur when large numbers of particles are observed under exactly similar circumstances, as for example in electron diffraction. ...........