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Вестник Забайкальского государственного университета

О влиянии инертности частицы на ее волновое представление

Попов Игорь Павлович

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

It is noted that for a monochromatic de Broglie wavelength corresponding to the free inertial particle wave group consists of a single wave, so the group velocity is identically equal to the phase. Justified by the presence in the expression for the wave kinetic energy factor ½, which is the same as in classical case which is due to the inertia of the particle. The representation of particle as a set of sub-particles, of which N is infinite, and therefore receive a confirmation of the equality group and phase velocities is described. It is shown that with the ratio for wave kinetic energy, expression for the total energy of the wave, the wave function and the Schrodinger equation for a free particle are inextricably linked

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It is noted that for a monochromatic de Broglie wavelength corresponding to the free inertial particle wave group consists of a single wave, so the group velocity is identically equal to the phase. Justified by the presence in the expression for the wave kinetic energy factor ½, which is the same as in classical case which is due to the inertia of the particle. The representation of particle as a set of sub-particles, of which N is infinite, and therefore receive a confirmation of the equality group and phase velocities is described. It is shown that with the ratio for wave kinetic energy, expression for the total energy of the wave, the wave function and the Schrodinger equation for a free particle are inextricably linked
Based on the equation of the free movement of inert body built formal analogue of the wave function, which is the solution of the differential equation, almost identical to the Schrodinger equation for a free particle. It is shown that the inverse image of the wave function does not substantially coincides with the equation of rectilinear uniform motion. From a comparison of the formal analogue of the wave function and the classical wave equation to obtain an expression for the wave kinetic energy of an inert body. Show that a formal analogue of the wave function is a function monochrome, for which the phase and group velocities coincide and are equal velocity of the body. We consider two alternative variants of formation of the wave packet for a formal analogue of the wave function, according to which the body either moves simultaneously with all measurement speed range, or with a single.
For the wave-particle duality of microscopic particles,the relationship of the velocity and the velocity of phase and the velocity of group for the microscopic particles is analyzed,it is pointed out that the velocity relating to the frequency and wavelength of two De Broglie expressions is not the velocity of group but the velocity of phase.On the contrary,the velocity in the defining expression of momentum is the real speed of particles,which doesn't equal to the velocity of phase but equals to the velocity of group.
In the first section of this paper I derive the formulas for the phase velocity and group velocity as a function of the total relativistic energy and the momentum of a particle. In the second section I derive similar formulas as a function of the de Broglie and the Compton wavelengths of the particle. In the third section, I derive similar formulas as a function of the angular frequency and the wave number. Finally, two additional meanings of the Compton wavelength are derived. The first one is derived from the equation of the group velocity in terms of the de Broglie and the Compton wavelengths and the second one from the range of the weak nuclear force.
In this article, it is shown that if a particle is at rest and has some harmonic oscillation, then a traveling particle with a constant speed corresponds to a De Broglie Wave.In 1924, De Broglie proposed that particles of matter exhibit wave like behavior, the expression for which became fundamental in the creation of wave mechanics (quantum mechanics).The main equation for this is the Schrödinger equation, which in the case of free motion of a particle becomes the expression for the De Broglie wave.This hypothesis is supported by many early experiments on the diffraction of electrons, and later the development of modern physics.However until now, the origin of this hypothesis remains unclear.We show that the expression for the De Broglie wavelength can be obtained from a more general well-known concept, that any particle at rest has a corresponding harmonic oscillation that transforms into a wave of a moving particle with the help of relativistic change of coordinates (the Lorentz transformations).In this way, a De Broglie wave can be explained as a spacetime wave.
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 energy of an element in a wave motion can be of two kinds: first, non-interactive like the energy of a set of independent pendulum bobs, and second, interactive like the potential energy of an element of a stretched string which is dependent only on the relative position of the neighboring elements. In the former case the group velocity is zero, and in the latter it equals the wave velocity. The slowing down of energy transmission in intermediate cases is discussed qualitatively and the ratio of group to wave velocity is calculated quantitatively in a number of cases on the assumption that the energy propagation is given not only by the product of energy density and group velocity, but also by the product of twice the interactive energy density and wave velocity. A general relation is suggested without proof for the connection between energy types and the ratio of group to wave velocity. The same ideas are applied to de Broglie's phase waves, the group velocity of which is the particle velocity.