Sunday, 16 February 2020

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:


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


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