Energy as rate-of-phase-change

[jjustinn]

I came across this very intriguing statement in another thread:

...you...may be interested to know, the fundamental relationship between phase and energy in quantum mechanics. Energy is the rate of change of phase. For instance, when the "phase clock" of a photon turns at a given rate, that is because the photon has a certain energy that is proportional to this rate. (The proportionality factor is h-bar. You can verify that multiplying h-bar by a measure of rotation rate like Hz gives you units of energy). Electrons, too, and indeed all particles, have a phase clock. Just as with a photon, an electron's energy is how fast its phase clock is turning.

I vaguely recall seeing this elsewhere, and there is something extremely seductive about it -- if nothing else, its ratio of simplicity to explanatory / predictive power is pretty massive.

However, I'm having trouble with the interpretation / ramifications, and I think it comes back to uncertainty relations / Fourier duality -- and maybe also wave/particle duality.

For instance, even the statement that there is a "rate of change of phase" poses some difficulty -- because what is it that "has" this phase? I suppose if you took the integral of a wavefunction's energy over any region at time t=a, you could say that this is the instantaneous rate of change of that wavefunction's "phase in that region", but can you even define "phase" in a region -- or for that matter, at an instant? I suppose you can take it a step further and integrate over a 4-volume, and then somehow have an "average" phase / energy over that 4-volume, but being an average, the value it would necessarily be less precise the larger the volume (and therefore, the more 'confident' you can be of the value)?

Or am I totally over-complicating this?

 

[KWillets]

This is basically one aspect of de Broglie. He postulated (extending from photons) that E is proportional to an intrinsic frequency. As mass increases relativistically with velocity, frequency increases. That rule is quite simple.

The trick is to get this all to work with SR, since a moving clock runs slower -- phase velocity has to increase, beyond c, to compensate. (Read his thesis for a better explanation.)

You may be overcomplicating. I believe the intended meaning is that "rate of change of phase" is frequency, which has to be measured over an extended waveform, with operators, etc.