Question

Why is the phase velocity of de broglie wave geater than speed of light in vacuum?

Answer 1

We can show that the phase velocity is greater than the speed of light by starting with the equation relating energy to momentum and rest mass:

E2=(pc)2+(m0c2)2

Now take the equations of quantum physics:

E=hf

and

p=hλ

Combine these equations, and substitute for the phase velocity:

vp=fλ

and with algebra you will get

v2p=c2+m20c4h2λ2

Since the second term on the right side is a square, it must be positive. That means that vp≥c, except, of course, for massless particles such as photons.

To find the group velocity, calculate vg=dωdk, where ω is 2πf and k is 2πλ. You'll find that it is less than the speed of light.

Not only that, but the answer is so interesting that I'll give it here. You can show that the group velocity is equal to the classical velocity:

vg=pγm0=pm=v

where γ=11−β2√.

If you know some calculus, you can readily prove this. (Hint: E=γm0c2.)