# How will you show that the velocity of matter wave is always greater than that of light?

##### 1 Answer

The energy

and its momentum by the relation

where

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

where

where

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

where

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.