Question

What do you mean by group and phase velocity? Show that the de-Broglie group velocity associated with the wave packet is equal to the velocity of the particle.

Answer 1

GROUP VELOCITY:-

When a number of waves of slightly different wavelengths and velocities travel together in a medium the observed velocity of this group of waves is called the Group velocity. Such a group of waves is called a wave packet.

PHASE VELOCITY:-

The velocity with which a wave travels through a medium is known as phase velocity or wave velocity.

RELATION BETWEEN PHASE AND GROUP VELOCITY.
Consider a particle of rest mass m_o  moving with a velocity v, which is very large and comparable
to c with v<c , its mass is given by the relativistic formula. 

`m = (m_0)/(sqrt(1)-(v^2/c^2)`

ω = 2πϑ …….. (ω= angular frequency)

` = 2π(E/h) = 2π ((mc^2)/h)`

And    `k = (2π)/λ = (2pπ)/h = (2π)/h(mv)`

Wave velocity is the phase velocity given as

`V_p = ω/K = c^2/V`

`V_p = (dω)/(dK)`

`v_g = ((dω)/(dv))/((dk)/(dv))`

`v_g = v`

This shows that a matter particles in motion is equivalent to a packet moving with group
velocity Vg whereas the component waves moves with phase velocity V_p .
Hence the relation between phase velocity and group velocity is:-

∴`v_p v_g = c^2`