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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?

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I'm not really sure what the question is. First of all, frequency of what? As for energy that isn't proportional to frequency, how about potential energy? – jld Mar 5 '13 at 19:50
The difference is the immense scale difference. Each particle in the cat has an attributed de Broglie wavelength, but since Planck's constant is so tiny these waves are very very small compared to the total size of the cat. – jld Mar 5 '13 at 19:57
In other words, a cat does have a fantastically large frequency. – emarti Mar 5 '13 at 20:26
This is getting a bit off the subject, but an object as large as a cat will decohere with any measuring system too fast for you to observe any quantum properties such as it's de Broglie wavelength. The wavelength/frequency of a cat is not observable, not even in principle. – John Rennie Mar 6 '13 at 7:51
I agree that X is much easier to measure than Y because, for instance, it has a lower mass (X=atom, Y=cat). But you should be careful about saying that something "cannot be measured in principle", which means that it is impossible to elucidate the desired quantity from the experiment on theoretical grounds. Such a statement requires a qualitative, not quantitative, difference. It was long thought that atomic and molecular matter-wave interference would be impossible. I agree that we shouldn't expect a cat-ter wave interferometer any time soon. – emarti Mar 16 '13 at 22:57
up vote 1 down vote accepted

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.

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