Show that group velocity of matter waves associated with a particle is equal to

the particle velocity(V_{group}=V_{particle})

#### Solution

Consider a particle of rest mass m, 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= mo /√1-(v^{2}/c^{2}) ............... (1)

Let ω be the angular frequency and k be the wave number of the de Broglie wave associated

with the particle. Here vis the frequency and A is the wavelength of the matter wave. Hence,it can be written that

ω = 2πv = 2( mc^{2} /h)

ω = 2π /h. m_{o}c^{2}/√1-(v^{2} / c^{2}). ........... (2)

k = 2π/λ = 2πp/h = 2πmv/h

k = 2π/h • m_{o}v/√1 - v^{2} /c^{2}

• The wave velocity is the the phase velocity given by

Vp = ω/k = c^{2}v.................. (3}

• Since the wave packet is composed of waves of slightly different wavelength and velocities, the group velocity is written as

v_{g} = dω/dk

• This can be calculated using equation (1) and (2) as

v_{g }=dω / dv /dk /dv

=[d/dv(c2 /√1-(v^{2} /c^{2}))][d/dv (v/√1-(v^{2} /c^{2})]^{1}

v_{g} =V

This shows that a matter particle in motion is equivalent to a wave packet moving with group velocity v_{g} whereas the component waves move with phase velocity, Vp.