Uncertainty principle, maybe it doesn't forbid simultaneous measurements of position and momentum?

Question

Consider the single-hole diffraction of an electron. We can make the hole as small as we like and determine the electron's position with arbitrary accuracy. When it is in the hole, we can hit it with a low-energy photon, and measure its momentum.

There is no guarantee that the photon will strike the electron. But it "could". It seems that we can measure the position and momentum simultaneously if we are lucky. Does it mean UP has nothing to do with simultaneous measurements but only with statistical uncertainty?

Answer 1

For the photon to resolve the hole, it has to have a wavelength smaller than the hole, which gives it a momentum on the order of the momentum uncertainty caused by the hole, defeating the purpose of a low energy photon in the first place.

Answer 2

Long story short: No, unfortunately not because you would measure random momentum with each new measurement.

Kudos for your curiosity and your Gedankenexperiment though.

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The idea behind your measurement might be that you measure the momentum once via observation with a photon and then you know the momentum for each electron in a consecutive measurement. I hope that I got your point right.

In this case you would nevertheless not be able to infer information from your measurement because at the time of measurement the electron would have arbitrary momentum.

So assume a measurement is down at time $t_1$ resulting in momentum $p_1$. The next electron you measure at $t_2$ with momentum $p_2$ you could not causate latter momentum with former momentum as the difference in momentum is arbitrary due to HUP.

So even if you could do those measurements you would not be able to gather some causally linked information from this other than HUP meaning your measured momenta were to be distributed.

But, and this has to be stressed, you could not perform any predictions on the momenta of electrons going through the hole that you described.

In fact HUP is even more general than just momentum and position.

It is a result from the Cauchy Schwarz inequality. The Cauchy Schwarz inequality basically states that the projection of one vector upon another vector is less than the product of the individual lengths of the vectors.

But this means that a formulation analogous to HUP for momentum an d position can be found for many pairs of variables. These pairs in turn are called conjugated variables(wiki).

To answer your question: No, it is not possible to measure $\underline{\text{the momentum}} $ in your Gedankenexperiment in a way that the data you collect can be interpreted apart from random fluctuations.