I would imagine a black hole of sub-atomic size would have negligible gravity compared to other influences it could exert on its environment, namely in the form of Hawking radiation. My guess is the electron would have a much higher probability of being scattered by high-energy photons, but I could be wrong. At any rate, a mountain-massed mini black hole has a temperature on the order of 10[sup]11-12[/sup] kelvin. It would seem such an energetic object ought to blow an atom apart long before it it felt the black hole’s gravity.
More accurately, it says that there is a lower bound on the product of errors in measuring conjugate variables, such as position and momentum in the same direction.
As for whether it’s a quantum experiment: unquestionably yes. Further, it’s a quantum gravity experiment, and there’s still no good theory of that out there.
Actually, I don’t think that would be a quantum gravity experiment. If we suppose that the black hole has a diameter a little less than the Bohr radius (thus making it ambiguous whether the electron hits it), it would be well above the Planck mass. The electron would, of course, be quantum mechanical, but that’s just a problem of quantum mechanics in a curved spacetime, which is fairly straightforward. It’d basically be the same calculation as for Hawking radiation, but in reverse.
Incidentally, I get that the total lifespan of such a hole would be on the order of a few days, and it’s going to be radiating something fierce. It would probably emit more electrons than it could absorb.