Average velocity of free particle

[mihailo]

Homework Statement

The wave function is given as Ψ(x,t) = Ae^[i(k1x-ω1t)] + Ae^[i(k2x-ω2t)].
Show that particle average velocity Vav = ħ(k1+k2)/2m equals ω2-ω1/k2-k1.
Average momentum of the particle is Pav = ħ(k2+k1)/2.

Homework Equations

p = ħk
E=ħω
K = 1/2 * mV^2

The Attempt at a Solution

if we calculate |Ψ(x,t)|^2 = 2*A(1+cos((k2-k1)x-(ω2-ω1)t)) form V=ω/k we get ω2-ω1/k2-k1.
if we use average momentum Vav = Pav/m we get Vav = ħ(k1+k2)/2m.
my problem is to get mathematically form ħ(k1+k2)/2m to ω2-ω1/k2-k1

 

[PeroK]

What about using the relationship between energy and momentum?