How will you show that the velocity of matter wave is always greater than that of light?
and its momentum by the relation
De Broglie proposed that just as light has both wave-like and particle-like properties, electrons also have wave-like properties. He proposed the following relation for electrons
Above relationship is now known to hold for all types of matter and all matter exhibits properties of both particles and waves. De Broglie concluded that velocity of the particle should always be equal the group velocity of the corresponding wave. As such magnitude of the group velocity is equal to the particle's speed.
This means that
This relation holds good in case of Special Theory of Relativity as well.
Now we know that in quantum mechanics, particles behave as waves with complex phases as well. The phase velocity is equal to the product of the frequency multiplied by the wavelength. By the de Broglie hypothesis, we see that for particles
Using relativistic relations for energy and momentum, we have
Above expression can be rewritten as
According to theory of special relativity, speed of particle
Therefore, the phase velocity of matter waves always exceeds
The phase velocity of matter wave does not carry any meaningful information like momentum, velocity etc., it can be greater than the speed of light without violating the special theory of relativity.