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Journal of North China Institute of Water Conservancy and Hydroelectric Power

The Velocity and the Velocity of Phase and the Velocity of Group for Microscopic Particles

Yang Wu

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

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.

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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.
This chapter describes the dual nature of electron/de Broglie waves. de Broglie suggested that like light, an electron also behaves as both a particle and a wave. Based on this hypothesis, a relationship between wavelength of electrons (a particle) and momentum was developed. It opines that ‘wavelength of the particle is inversely proportional to momentum’. This is called the de Broglie relation. On the basis of the above facts, we have developed a relationship between wavelength and voltage applied and this relation may be stated as ‘the wavelength of a particle is inversely proportional to the square root of the voltage applied to the system’. This chapter also uses the Davisson and Germer experiment to show the wave nature of the particle, i.e., diffraction of electrons by crystal lattice. The experimental results were the same as found in the case of the diffraction of X-ray by crystals. This chapter also provides an idea on quantisation of angular momentum. It also discusses Heisenberg’s uncertainty principle, phase velocity, group velocity, and uncertainty relation between energy and time. The chapter comprises the experimental evidence of Heisenberg’s uncertainty principle, such as by diffraction of electrons through a slit and gamma ray microscope thought experiment, which is shows the physical significance of the uncertainty principle. Sufficient references, solved problems, and questions on concepts have been provided at the end of chapter.
The Wave-Particle Duality is a basic property of microscopic particles. As a basic concept of quantum mechanics, the wave-particle duality theory from elementary particles to big molecules had been verified by lots of experiments. Different from electromagnetic wave, the matter wave’s propagation is not only fast but also adjustable. The group velocity with which the overall envelope shape of the wave , namely the related particle’s propagation and information convey speed is changeable with its wavelength. When the particle’s energy and wavelength, in its non-dispersive propagation, reach to definite values, the group velocity can turn to very high and perhaps is possible to exceed over the light speed in vacuum. Take electron as an example, if the free electron beam gains energy higher than around 4.094×10ˉᴵ⁴J and wavelength shorter than around 2.426×10ˉᴲnm, the group velocity could surpass the light speed in vacuum. According to the special relativity theory, the light speed in vacuum is the upper limit in the Universe and no matter can exceed over that. As the de Broglie hypothesis and the deduction in this paper do not take relativity theory as premise, the conclusion in this paper should be universality. Wish this paper could provide a different viewpoint for exploration of some scientific problems, such as Faster-Than-Light movement, quantum entanglement mechanism and so on
This article points out a falsehood in some documents about the relations between the velosity of singlw phase and group phase of de Broglie wave,and explains that the superlight velosity of wave phase is not contracted with wave-particle duality.The article makes a further study of diffusion of de broglie wave packet and comes to a cvonclusion that the smoothly changing wave packet of free particle and frequency spectrum functionφ(k) is basically not diffusing,on the contrary,it diffuses faster and apparently.
The Wave-Particle Duality is a basic property of microscopic particles. As a basic concept of quantum mechanics, the wave-particle duality theory from elementary particles to big molecules had been verified by lots of experiments. Different from electromagnetic wave, the matter wave’s propagation is not only fast but also adjustable. The group velocity with which the overall envelope shape of the wave , namely the related particle’s propagation and information convey speed is changeable with its wavelength. When the particle’s energy and wavelength, in its non-dispersive propagation, reach to definite values, the group velocity can turn to very high and perhaps is possible to exceed over the light speed in vacuum. Take electron as an example, if the free electron beam gains energy higher than around 4.094×10ˉᴵ⁴J and wavelength shorter than around 2.426×10ˉᴲnm, the group velocity could surpass the light speed in vacuum. According to the special relativity theory, the light speed in vacuum is the upper limit in the Universe and no matter can exceed over that. As the de Broglie hypothesis and the deduction in this paper do not take relativity theory as premise, the conclusion in this paper should be universality. Wish this paper could provide a different viewpoint for exploration of some scientific problems, such as Faster-Than-Light movement, quantum entanglement mechanism and so on