The group velocity of matter waves associated with a moving particle is:

Concept:

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes known as the modulation or envelope of the wave propagates through space.

The group velocity is defined by the equation:

\({v_g} = \frac{{d\omega }}{{dk}}\)

Where ω = wave’s angular frequency

k = angular wave number = 2π/λ 

Wave theory tells us that a wave carries its energy with the group velocity. For matter waves, this group velocity is the velocity u of the particle.

Energy of a photon is given by the planck as:

E = hν

With ω = 2πν

ω = 2πE/h      ----- (1)

Wave number is given by:

k = 2π/λ = 2πp/h    ----(2)

where λ = h/p (de broglie)

Now from equations 1 and 2, we get:

\(d\omega = \frac{{2\pi }}{h}dE;\)

\(dk = \frac{{2\pi }}{h}dp;\)

\(\frac{{d\omega }}{{dk}} = \frac{{dE}}{{dp}}\)

By definition: \({v_g} = \frac{{d\omega }}{{dk}}\)

v_g = dE/dp   ---- (3)

If a particle of mass m is moving with a velocity v, then

\(E = \frac{1}{2}m{v^2} = \frac{{{p^2}}}{{2m}}\)

\(\frac{{dE}}{{dp}} = \frac{p}{m} = {v_p}\)    ---(4)

Now from equations 3 and 4:

v_g = v_p