I understand that after passing through a narrow slit a photon's momentum is uncertain, however since its position can be inferred from having passed through the slit, a subsequent position measurement fully constrains the momentum direction. Therefore as long as we know the photon's wavelength, we have simultaneously measured x and p to arbitrary accuracy.

I forsee two possible responses:

1) Even though we may start with a photon of known $\lambda$, passing through the slit makes $\lambda$ uncertain. However I've never seen this discussed; always diffraction is described as changing momentum direction, not magnitude.

2) The HUP says that a position measurement makes subsequent momentum measurements uncertain, NOT that position and momentum cannot be simultaneously determined. However this is in strong tension with every description of the HUP I've ever seen, including an answer to a previous similar question here. Note that the previous link might be considered a duplicate of this question, however, I think both that question and answer are not focused enough to be informative.

  • $\begingroup$ Please note that each photon in the oicture here physics.stackexchange.com/questions/468533/are-photons-blinking/… leaves a dot which has a delta(x). The delta(p) can easily be fulfilled which you can calculate yourself as the dots are of micron size, and the wavepacket of the frequency of the photon can easily be accomodated. $\endgroup$ – anna v Mar 26 at 18:08
  • $\begingroup$ @annav, your argument would only hold if we couldn't arbitrarily increase the distance between the slit and the screen, thus being able to arbitrarily reduce the angular uncertainty that results from your delta(x) $\endgroup$ – user1247 Mar 26 at 19:05
  • $\begingroup$ The color of the photons gives the momentum, the point is the measurement on the screen. That is the HUP., at the point of measurement. $\endgroup$ – anna v Mar 26 at 19:11

The uncertainty principle is a statement about the physical state of a system at one particular time, not about its past evolution. After the position measurement you mention, the location of the particle has been fixed but its direction of travel after this measurement is unknown.

As for the direction of travel between the first slit and the second measurement, with the benefit of hindsight it is entirely reasonable to suppose the particle went straight from one slit to the later measured position, but this doesn't tell you what the state of motion was as it left the slit! After all, it could have been expanding out of that slit in all directions, and probably was. The situation is loosely comparable to classical diffraction of water waves. If my detector says the water wave arrives here, that doesn't necessarily mean the water wave didn't also have some non-negligible amplitude at lots of other places. Thus the position measurement does not pin down what the state was before the measurement nearly as much as you guessed.

Of course these things were thought through at length by the early pioneers of quantum theory, and the type of observation you describe has been repeatedly discussed.

When in doubt about the content of the Heisenberg Uncertainty Principle, here is a good guide: fall back on your knowledge of Fourier analysis. A pulse of some wave-like disturbance having finite length will have a non-zero spread in its frequency composition. It cannot avoid it; it is a mathematical certainty based on the very defintion of what frequency and wavelength are. That is what the HUP is all about. The only quantum feature is that the wavelength (a position-related quantity) now relates to the momentum. That is the aspect which classical mechanics could not have told you.

  • $\begingroup$ I'm struggling to understand the tension between your dismissal of "benefit of hindsight analysis" and what I think are traditional forms of inferential "measurement." For example when measuring position at a slit, we are similarly forming an analysis on "benefit of hindsight" that at some previous time the particle must have been at the slit, because we received some signal to that effect. In light of this, I fail to see how the inference regarding momentum is in this case fundamentally any different! I understand the fourier x-k uncertainty principle in classical wave theory. $\endgroup$ – user1247 Mar 26 at 19:02
  • $\begingroup$ @user1247 the "wave" in the formalism is a probability wave. An ensemble of measurements will show the wave nature. $\endgroup$ – anna v Mar 26 at 19:12
  • $\begingroup$ @annav yes I know... $\endgroup$ – user1247 Mar 26 at 19:23

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