Show that the velocity of a moving material particle is equivalent to the velocity of wave packet.
Answers
The energy
E
of photon is given by the Planck–Einstein relation
E
=
h
ν
and its momentum by the relation
p
=
E
c
=
h
λ
where
ν
and
λ
denote the frequency and wavelength of the light,
c
the speed of light, and
h
is Planck’s constant.
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
λ
=
h
p
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
v
g
=
∂
E
∂
p
where
E
is the total energy of the particle, and
p
its momentum. For non-relativistic cases it follows that
v
g
=
∂
(
1
2
p
2
m
)
∂
p
⇒
v
g
=
p
m
=
v
where
m
and
v
are mass and velocity of the particle respectively.
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
v
p
=
ω
k
=
E
ℏ
p
ℏ
=
E
p
.
Using relativistic relations for energy and momentum, we have
v
p
=
E
p
=
γ
m
0
c
2
γ
m
0
v
=
c
2
v
where
E
is the total energy of the particle,
γ
is the Lorentz factor, and
c
is already defined above as speed of light.
Above expression can be rewritten as
v
p
×
v
=
c
2
According to theory of special relativity, speed of particle
v
<
c
for any particle that has mass.
⇒
v
p
>
c
Therefore, the phase velocity of matter waves always exceeds
c
.
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.