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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?

Qmechanic
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asim dahal
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    I will clarify that the HUP doesn't just forbid simultaneous measurement of position and momentum. It states something far stronger -- that systems simply do not have a precise momentum and a position at the same time. The HUP has absolutely nothing to do with measurement. – Prahar Aug 26 '23 at 11:52
  • The uncertainity principle as usually stated, the Kennard--Weyl relation, is statistical in nature. As for simultaneous measurements, the situation is different. See https://en.wikipedia.org/wiki/Uncertainty_principle#Additional_uncertainty_relations. There are question discussing this on this site as well. I'll link them after I find them. – Apoorv Potnis Aug 26 '23 at 12:52
  • Related: https://physics.stackexchange.com/a/623184/81224 and https://physics.stackexchange.com/a/169736/81224. – Apoorv Potnis Aug 26 '23 at 12:59
  • @Prahar Whether the systems actually possess simultaneous position and momentum is an interpretation dependent statement. You should instead say that the wavefunction of the system cannot simultaneously be an eigenstate of position and momentum. HUP is a theorem about the wavefunction's standard deviations in position and momentum. – Ryder Rude Aug 26 '23 at 13:00
  • @Prahar https://physics.stackexchange.com/a/600015/81224. From that answer "There is a sense in which a simultaneous and arbitrarily precise measurement of position and momentum is not only possible, but also routinely made in many quantum labs, for example quantum-optics labs. Such measurement is indeed at the core of modern quantum applications such as quantum-key distribution." – Apoorv Potnis Aug 26 '23 at 13:08
  • @Prahar If you can measure position and momentum precisely, then in what sense does the particle not have those properties? Do you mean the momentum could have been anything in my example above? – asim dahal Aug 26 '23 at 14:19
  • @RyderRude - I'm curious what interpretation states otherwise? That a wavefunction possesses certain properties but the system does not? – Prahar Aug 26 '23 at 16:55
  • @asimdahal - The uncertainties that the HUP talks about is regarding certain properties of the wavefunction as RyderRude said. No wavefunction has a precise value of momentum and position at the same time. The HUP is a statement about the fundamental NATURE of quantum objects -- not about the process of measurement. – Prahar Aug 26 '23 at 16:59
  • The point is that due to this fundamental property of position and momentum, it follows as a corollary that they also cannot be measured simultaneously. The latter is how the HUP is typically phrased in pop sci writings. I only bring this up here because in your post you are trying to get around HUP by modifying your measurement procedure. What I am saying that no amount of change in the measuring procedure will help you here - because of the lack of simultaneity in position and momentum is a property of the system/wavefunction. – Prahar Aug 26 '23 at 17:02
  • @Prahar"No wavefunction has a precise value of momentum and position at the same time," True. The wavefunction changes after measurement, and you could argue the measured value of momentum is not the property of previous wavefunction. I would argue that we still got the precise values for position and momentum of the electron. I am trying to say that instead of phrasing UP in terms of simultaneous measurement, maybe it is better to regard it as a statistical principle. The usual one presented in textbooks in terms of standard deviations, which requires an ensemble. – – asim dahal Aug 27 '23 at 05:13
  • @Prahar For instance, hidden variable interpretations take the wavefunction to be an incomplete description of the system. In these interpretations, the particle possesses a well defined position and velocity but the wavefunction does not. – Ryder Rude Aug 27 '23 at 10:37

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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.

JEB
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  • "For the photon to resolve the hole". Is there no way that the photon interacts with the electron? Doesn't the interaction happen at a point? – asim dahal Aug 27 '23 at 05:09
  • What makes you think the interaction is at a point? – JEB Aug 27 '23 at 14:29
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Long story short: No, unfortunately not because you would measure random momentum with each new measurement.

Kudos for your curiosity and your Gedankenexperiment though.

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.

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

    Yes, that's what I mean by "statistical uncertainty".

     – asim dahal Aug 26 '23 at 14:33