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For example, one could change it to be so high that the energy associated with the electron is sufficient to cause a black hole to form. Since we don't see electrons spontaneously turning into black holes, doesn't this imply some practical upper limit?

Also, we can choose the potential so that the phase frequency is zero. In that case, wouldn't interference effects go away? But we don't see that happening either, I think.

So the question is, does a real electron have a single definite quantum phase frequency? Or is that just something that has no physical reality at all, and can be assumed to be anything you want?

(Note that I am NOT asking about absolute quantum phase, just the frequency. It's easy to prove that one can add constants to potentials in classical mechanics and classical EM with zero effect, so I accept that; the question is about to what extent that is also true in QM. The Aharonov-Bohm effect shows that potentials can have SOME effect in QM, but perhaps only in ways that are themselves gauge-invariant. So my question could also be viewed as being about the limits (if any) of such invariance.)