The concept of Wave packets and group velocities;

De-Broglie Wave Velocity;

The De-Broglie waves associated with the particle are expected to move with the same velocity with which the particle is moving. However, the velocity of the De-Broglie waves is different from the velocity of the particle. If vₚ is the velocity of the De-Broglie waves, then

vₚ=𝛎𝝀 ....(1)

Where 𝛎 is the frequency of waves and 𝝀 is the wavelength.

But 𝝀=h/mv ....(2)

Also 𝛎=E/h ....(3)

Where E is the energy of the quantum of waves and h is Planck's constant.

Since, E=mc²

Therefore, from eqn.(3),  𝛎=mc²/h ....(4)

Using equation (2) and (4) in eqn.(1), we get

vₚ=mc²/h×h/mv=c²/v

or vₚ=(c/v)c ....(5)

Since c >> v, it is clear from eqn.(5) that the De-Broglie wave velocity ( also called as the phase velocity) is greater than the velocity of light.

vₚ > c

According to theory of special relativity, speed of particle vₚ < c for any particle that has mass.

The phase velocity of matter waves always exceeds c,which is not possible. So a particle cannot be represented by a single harmonic wave.

Group velocity and Wave packet;

It is not a single wave but a group of waves which propagates in association with the motion of the particle.

The velocity with which the group of waves travel in association with the particle is called the group velocity. The group of waves is also termed as the wave packet or wave group.

Wave packet

A group of waves propagates in association with the motion of the particle. This group of waves is also termed as the wave packet or wave group. Thus a wave packet refers to the case where two or more than two waves exist simultaneously.

Group velocity

The velocity with which the group of waves (wave packet) travels in association with the particle is called the group velocity (v𝚐).

Also, v𝚐= v i.e the particle velocity is equal to the group velocity of the wave associated with it.

Group velocity and phase velocity.

Relation between phase velocity and group velocity of matter waves.

We know that De-Broglie wave velocity (also called as the phase velocity) is given by

vₚ= c²/v

But it has been shown that the particle velocity is equal to the group velocity of the wave associated with it i.e v𝚐= v

vₚ= c²/v𝚐

vₚ.v𝚐 = c² ....(6)

Equation (6) gives the relation between phase velocity and group velocity of matter waves.

Questions;

Q1) What is wave packet?

Ans)A group of waves propagates in association with the motion of the particle. This group of waves is also termed as the wave packet or wave group. Thus a wave packet refers to the case where two or more than two waves exist simultaneously.

Q2) What is group velocity?

Ans)The velocity with which the group of waves (wave packet) travels in association with the particle is called the group velocity (v𝚐).

Q3) What is the relation between phase velocity and group velocity of matter waves?

Ans)We know that De-Broglie wave velocity (also called as the phase velocity) is given by

vₚ= c²/v

But it has been shown that the particle velocity is equal to the group velocity of the wave associated with it i.e v𝚐= v

vₚ= c²/v𝚐

vₚ.v𝚐 = c² ....(6)

Equation (6) gives the relation between phase velocity and group velocity of matter waves.

Q4) What is De-Broglie Wave Velocity?

Ans)vₚ=(c/v)c ....(5)

Since c >> v, it is clear from eqn.(5) that the De-Broglie wave velocity ( also called as the phase velocity) is greater than the velocity of light.