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in vacuum, the angular frequency is proportional to the wavenumber:

This is a linear dispersion relation, in which case the waves are said to be non-dispersive. That is, the phase velocity and the group velocity are the same:

and thus both are equal to the speed of light in vacuum, which is frequency-independent.

De Broglie dispersion relations

For de Broglie matter waves the frequency dispersion relation is non-linear:

The

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