Question

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

Answer 1

Hint : The given question is a conceptual question. In order to write the answer for it we need to know all the details about the matter wave and speed of light. Then only we can conclude with the correct explanation for the given question. Also, we need to know the relation between phase velocity of matter wave and de Broglie hypothesis.

Complete step by step answer
Step one
First of all let us define matter waves.
We know this fact that in quantum mechanics, particles behave as waves. So, all matters behave like a wave.
Let us assume the phase velocity to be ${v_p}$
Step three
From de Broglie hypothesis we can write it as,
${v_p} = \dfrac{\omega }{k}$
$ \Rightarrow {v_p} = \dfrac{E}{h} \times \dfrac{h}{p} = \dfrac{E}{p}$…………………(i)
Step two
Now we need to relate phase velocity with the relation of energy and momentum.
Therefore, we can write equation (i) as
${v_p} = \dfrac{E}{p} = \dfrac{{m{c^2}}}{{mv}} = \dfrac{{{c^2}}}{v} = \dfrac{c}{\beta }$
$\beta $is the speed as a fraction of the speed of light.
Now, as the particle speed $v < c$ for any particle which has mass, the phase velocity of matter waves will exceed.
It means,${v_p} > c$

So, we can see that the velocity of the matter wave is more than that of light.

Note: We should also know about Quantum mechanics. Quantum mechanics is a branch of science which deals with the behavior of matter and light on atomic and subatomic scale. According to de Broglie matter behaves both like a particle and a wave. So, according to him every moving particle will have its wavelength. It means both the microscopic and macroscopic matter exhibits the wave nature.