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Show that Group Velocity of Matter Waves Associated with a Particle is Equal to the Particle Velocity(Vgroup=Vparticle) - Applied Physics 1

Answer in Brief

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

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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-(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. moc2/√1-(v2 / c2). ........... (2)         

k = 2π/λ = 2πp/h = 2πmv/h
k = 2π/h • mov/√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
vg = dω/dk
• This can be calculated using equation (1) and (2) as
vg =dω / dv /dk /dv
=[d/dv(c2 /√1-(v2 /c2))][d/dv (v/√1-(v2 /c2)]1

vg =V

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

Concept: Phase Velocity and Group Velocity
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