PHGN462: Advanced E&M
Quiz 3 – September 28, 2009
NAME:
1. Dispersion relations apply to quantum mechanical waves as well as electromagnetic waves.
a. A free non relativistic particle of mass, m, is described by the Schroedinger equation:
−
∂ψ(~r, t)
h̄2 2
∇ ψ(~r, t) = ıh̄
,
2m
∂t
~
which is satisfied by the plane wave function ψ(~r, t) = Aeı(k·~r−ωt) . Find the phase velocity, vp =
vg = ∂ω
∂k , for this wave in terms of the momentum, p = h̄k. Show all work.
(1)
ω
k,
and group velocity,
b. A free relativistic particle of mass, m, is described by the Klein-Gordon equation:
(−c2 h̄2 ∇2 + (mc2 )2 )ψ(~r, t) = −h̄2
~
∂2
ψ(~r, t)
∂t2
which is satisfied by the plane wave function ψ(~r, t) = Aeı(k·~r−ωt) . Find the phase velocity, vp =
vg = ∂ω
∂k , for the relativistic particle in terms of the momentum, p = h̄k.
1
(2)
ω
k,
and group velocity,