Is energy always proportional to frequency? Google has no results found for "energy not proportional to frequency" and many results for E=hf. Is there an example of an energy that is not proportional to frequency?
 A: Yes. For photons in vacuum, the energy per photon is proportional to the photon's classical, electromagnetic frequency, as $E = \hbar \omega = h f$. Here, we see a connection between two classical properties of light: the energy and frequency.
What is surprising is that the relation holds for matter, where there is no classical equivalent of the frequency. Nevertheless, in an interferometry experiment, an relative energy shift of $\Delta E$ can lead to an observable frequency difference $\Delta f$, so that the phase of an interferometer operated for a time $T$ is $\phi = \Delta E\,T/\hbar$. This was originally observed in neutrons and has more recently been seen in electrons and atoms. Even the rest mass energy $mc^2$ has a equivalent frequency, which is known as the Compton frequency $\omega_C = mc^2/\hbar$. While we can not (currently) experimentally measure it, it can be inferred from atom interferometry experiments.
The general idea of a matter-wave frequency occurs where it is possible to make and readout a superposition state, which does not occur classically.
