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What is the definition of energy $E$ given a dispersion relation $\omega=\omega(k)$ where $k=|\vec k|$ and $\omega$ is not necessarily linearly proportional to $k$? What about momentum $\vec p$?

This is in the context of quantum mechanics.

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Sorry, As lubos motl corrected It is Planks relationship and not Einsteins(and the missing 2pi factor). I think it will be applicable in a situations where the dispersion relation need is non linear. – Prathyush Mar 30 '13 at 13:54

$E = \hbar \omega$

It doesn't matter what the form of $\omega(\vec{k})$ is, whether it's linear or not, $E=\hbar\omega$.

e.g. For massive particles, $E = \frac{\hbar^2}{2m} |\vec{k}|^2$, which is parabolic, not linear, and $\omega = \frac{\hbar}{2m} |\vec{k}|^2$.

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