Matter Wave Phase
In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that
Using relativistic relations for energy and momentum, we have
where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum, the Lorentz factor, c the speed of light, and β the speed as a fraction of c. The variable v can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave. Since the particle speed for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.
and as we can see, it approaches c when the particle speed is in the relativistic range. The superluminal phase velocity does not violate special relativity, as it carries no information. See the article on signal velocity for details.
Read more about this topic: Phase Velocity
Famous quotes containing the words matter, wave and/or phase:
“I must beg you to indulge me in the matter of hyphens.... You will find that I have marked out a great many in the proofs. We are in danger of Germanizing our printing by using them so much, and I have a very decided preference in the matter.”
—Woodrow Wilson (1856–1924)
“The history of any nation follows an undulatory course. In the trough of the wave we find more or less complete anarchy; but the crest is not more or less complete Utopia, but only, at best, a tolerably humane, partially free and fairly just society that invariably carries within itself the seeds of its own decadence.”
—Aldous Huxley (1894–1963)
“The Indians feel that each stage is crucial and that the child should be allowed to dwell in each for the appropriate period of time so that every aspect of his being can evolve, just as a plant evolves in the proper time and sequence of the seasons. Otherwise, the child never has a chance to master himself in any one phase of his life.”
—Alan Quetone (20th century)