quantum mechanics - Why don't the De Broglie dispersion relation contain a constant term?

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:

$$ \omega = \frac{\hbar^2 k^3}{2m} \times \frac{1}{\hbar k} = \frac{\hbar k^2}{2m} $$