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:
“It is worth the while to live respectably unto ourselves. We can possibly get along with a neighbor, even with a bedfellow, whom we respect but very little; but as soon as it comes to this, that we do not respect ourselves, then we do not get along at all, no matter how much money we are paid for halting.”
—Henry David Thoreau (1817–1862)
“As the bird trims her to the gale,
I trim myself to the storm of time,
I man the rudder, reef the sail,
Obey the voice at eve obeyed in prime:
“Lowly faithful, banish fear,
Right onward drive unharmed;
The port, well worth the cruise, is near,
And every wave is charmed.””
—Ralph Waldo Emerson (1803–1882)
“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)