Information’s Super-Speed and Adjustable Convey with Matter Wave’s Non-Dispersive Propagation

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

According to the wave-particle duality theory, de Broglie combined the Einstein mass-energy equation E=mc²,one base of the relativity theory related to mass and energy, with the Planck relation E=hν ,one base of quantum mechanics theory related to wave frequency and energy, then equalized both to: mc²= hν . However, de Broglie considered that not all matter move with the velocity of light speed in vacuum, so he changed the mass-energy equation from E=mc² to E=mv² by replacing c with the actual velocity v. and equal both as mv²= hν. This important change means that the de Broglie hypothec did not take the relativity theory as its theoretical base and therefore the particle's moving velocity is not restricted. This newly de Broglie hypothec was formulated as below: mv²= hν (1) Formula (1), (2) also can be written as p=ћk Among formula (1),(2),(3)and(4), ν is particle velocity, ν is wave frequency, p is momentum, E is energy, λ is wavelength, k is angular wavenumber, ω is angular frequency, h is Planck constant and ћ is reduced Planck constant. The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes -known as the modulation or envelope of the wave propagates through space and is often thought of as the velocity at which energy or information is conveyed along a wave. The group velocity, vg is defined by equation ( This is the group velocity of light, namely the photon's speed in vacuum. 1.2) rest mass m≠0，particle's speed approaches to light speed in vacuum-ultrarelativistic limit, as pc ›› mc², so E=pc. thus

vg= =c
This is the group velocity of related particle, which moves with the light speed in vacuum. 1-3) rest mass m≠0 and particle's speed less than c in the nonrelativistic limit, In formula(8), p=mv. According to author's estimate, item mc² is less than item p² 2m by around two orders of magnitude. Whether or not to make approximate assumption to formula (8), as value of mc² is invariant, the result is the same as:

2.0) De Broglie's Propagation-Free Particle's Non Relativistic Propagation
De Broglie concluded that the velocity of a particle should always be equal to the group velocity of the corresponding wave. This conclusion can be proved as below: As the total energy for a free particle in non-relativistic propagation is

5-3
The formula (12) indicates that the group velocity vg is changeable with wavelength λ, and the value of vg is unlimited. When the wavelength λ reduces to certain extent, the group velocity, vg can increase much and even exceed over the light speed in vacuum. Suppose Formula(16),(17)indicate that when electron's wavelength is shorter than 2.426×10ˉᴲnm and electron's energy is higher than 4.094×10ˉᴵ⁴J, the electron's propagation speed(18)is exceed over the light speed in vacuum. This result means that the matter wave is quite different from the electromagnetic wave. Although both propagate information, the convey velocity for matter wave is adjustable and even able to faster than that of light in vacuum, while the convey velocity of electromagnetic wave is generally equal to the light speed and almost unchangeable.

Facts and Data
The electron microscope uses a beam of accelerated electrons under a very high voltage of several 10ᴲ to 3x10⁶ voltages (the HVEM can be applied to 200 to 1000 kV) between a cathode, as an electron source and an anode, as an acceleration part, going through a vacuum area to illuminate the specimen and create an image. As the wavelength of electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have higher resolving power than light microscopes and can reveal the ultrastructure of smaller objects.

Conclusion
Different from electromagnetic wave, the matter wave's propagation, that is, the related particle's movement and information's convey is not only fast but also adjustable. The velocity can be changed with its wavelength and In the non-dispersive propagation, it can reach to very high speed and is able to exceed over the light speed in vacuum. Take electron as an example, the electron's propagation and related information's convey speed could surpass the light speed in vacuum, if its wavelength is shorter than around 2.426×10ˉᴲnm and its energy is higher than around 4.094×10ˉᴵ⁴J in the non-dispersive propagation. As the de Broglie hypothesis did not take relativity theory as premise and the deduction in this paper also proceeds under non-relativity condition, the conclusion in this paper should be universality. According to Special Relativity Theory, the light speed in vacuum is the upper limit in the Universe and no matter can exceed over that. Wish this paper could provide a different viewpoint to understand some scientific problems such as Faster-Than-Light movement, quantum entanglement mechanism and so on.
The author: Wang Xinye graduated from Tongji University, Shanghai, China in 1964 and now is a retired senior engineer (in China) and a retired scientist (in Singapore)