Question

Show that group velocity of matter waves associated with a particle is equal to
the particle velocity(V_group=V_particle)

Answer 1

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-(v²/c²) ............... (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( mc² /h)
ω = 2π /h. m_oc²/√1-(v² / c²). ........... (2)         

k = 2π/λ = 2πp/h = 2πmv/h
k = 2π/h • m_ov/√1 - v² /c²
• The wave velocity is the the phase velocity given by    

Vp = ω/k = c²v.................. (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-(v² /c²))][d/dv (v/√1-(v² /c²)]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.