Wikipedia says that the dispersion relation for a non-relativistic particle is:
$$ \omega = \frac{\hbar k^2}{2m}. $$
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:
$$\omega = \frac{E k}{p}$$
Substituting the non-relativistic energy limit:
$$\omega = \frac{\left( m c^2 + \frac{p^2}{2m} \right)k}{p}$$
Substituting the momentum:
$$\omega = \frac{\left( m c^2 + \frac{\hbar^2 k^2}{2m} \right)}{\hbar }$$
Performing the division, I get:
$$\omega = \frac{m c^2}{\hbar} + \frac{\hbar k^2}{2m}$$
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:
$$ E = \frac{p^2}{2m}$$
Everything else follows from there:
$$ \omega = \frac{\hbar^2 k^3}{2m} \times \frac{1}{\hbar k} = \frac{\hbar k^2}{2m} $$
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