Wikipedia says that the dispersion relation for a non-relativistic particle is:
ω=ℏk22m.
But when I tried to calculate it myself, I seem to get a constant term in that formula. My derivation is the following:
Reordering the De Broglie relations I have a generic dispersion relation:
ω=Ekp
Substituting the non-relativistic energy limit:
ω=(mc2+p22m)kp
Substituting the momentum:
ω=(mc2+ℏ2k22m)ℏ
Performing the division, I get:
ω=mc2ℏ+ℏk22m
Maybe I miss something obvious. The relation in the Wikipedia doesn't contain that constant term why? Maybe in the non-relativistic case the mass energy is not considered as energy at all? That would be interesting...
Answer
I believe this is simpler than you make it to be. If you want to substitute in the non-relativistic energy relation, then the correct energy term is just the kinetic energy:
E=p22m
Everything else follows from there:
ω=ℏ2k32m×1ℏk=ℏk22m
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