Single slit diffraction and the HUP

[J O Linton]

TL;DR Summary

Does the fact that you can measure the lateral momentum of a photon passing through a single slit contradict the HUP?

In their articles on the Heisenberg Uncertainty Principle Wikipedia says "there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known"; Britannica says " the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory. The very concepts of exact position and exact velocity together, in fact, have no meaning in nature."

In a previous thread Single slit diffraction I suggested that the unpredictability in the trajectory of a single photon passing through a single slit could be explained by saying that the slit imposes limits on the position of the photon in the Y direction thereby introducing an uncertainty in the Y momentum causing the diffraction. Granted this, when the photon lands on the screen and its deviation is noted, the lateral momentum can easily be calculated. We now know both the position and momentum of the photon as it passed through the slit to an accuracy much greater than that allowed by the HUP. How is this consistent with the statements quoted above?

 

[sophiecentaur]

TL;DR Summary: Does the fact that you can measure the lateral momentum of a photon passing through a single slit contradict the HUP?

We now know both the position and momentum of the photon as it passed through the slit to an accuracy much greater than that allowed by the HUP.

Is that right? Where the photon arrives can be known to within the pitch of the sensor but how accurately can you tell the momentum change? By an assumed geometric path through the slit? Doesn't that assume that the photon has a geometrically straight path towards the slit? That's quite an assumption, imo. It has no 'location' during its journey from the source.

 

[Nugatory]

We now know both the position and momentum of the photon as it passed through the slit to an accuracy much greater than that allowed by the HUP.

We would know (could calculate) the transverse momentum if we could know the time when the particle passed through the slit. But we don’t.

 

[DrClaude]

The HUP is not about individual measurements on a single system. It is about the variance of the expectation value of an ensemble of identically prepared systems (i.e., in the same quantum state). Send a bunch of identically-prepared particles through the slit, and the diffraction pattern will follow the HUP.

 

[J O Linton]

Both the articles I quoted imply that the HUP applies to individual particles. Both Einstein and Bohr in their famous debates in the 1920's assumed that this was the case. Were they both wrong?

 

[PeterDonis]

Both the articles I quoted

Are not textbooks or peer-reviewed papers. And this is a subject where even a "reputable" source that isn't one of those, like the Encyclopedia Britannica, is not really a good source to learn from. You really need to work through the actual math and what it does and does not tell us, rather than relying on anyone's attempt to describe it in ordinary language.

Both Einstein and Bohr in their famous debates in the 1920's assumed that this was the case.

That's because it was the 1920s and QM was a very new theory and was not fully understood by anyone, even Einstein and Bohr. Now it's almost a century later and we have a lot more information than they did.

 

[J O Linton]

We would know (could calculate) the transverse momentum if we could know the time when the particle passed through the slit. But we don’t.

I am assuming that the transverse momentum is equal to $p sin \theta$ where $p = h/ \lambda$

 

[PeterDonis]

The HUP is not about individual measurements on a single system. It is about the variance of the expectation value of an ensemble of identically prepared systems (i.e., in the same quantum state).

I think this is actually interpretation dependent. On an ensemble interpretation, yes, what you say is what the interpretation would say. But on an interpretation where the quantum state describes the state of an individual system, the HUP is about possible states of an individual system: the fact that there is no quantum state in which both position and momentum are precisely defined becomes a restriction on possible states of individual systems, not just possible predictions about measurement results on ensembles.

 

[J O Linton]

I think this is actually interpretation dependent. On an ensemble interpretation, yes, what you say is what the interpretation would say. But on an interpretation where the quantum state describes the state of an individual system, the HUP is about possible states of an individual system: the fact that there is no quantum state in which both position and momentum are precisely defined becomes a restriction on possible states of individual systems, not just possible predictions about measurement results on ensembles.

Would it, therefore, be fair to say that, on an interpretation which allows objective reduction (or collapse) of the wave function, precise knowledge of both x and p is only possible after the collapse, not before?

 

[PeterDonis]

Would it, therefore, be fair to say that, on an interpretation which allows objective reduction (or collapse) of the wave function, precise knowledge of both x and p is only possible after the collapse, not before?

No. The quantum state of the system will never be one in which both x an p are precisely known--because there is no such quantum state at all. No such state exists. Neither before nor after the collapse.

 

[Morbert]

A relevant paper by Aharonov and Vaidman: https://www.tau.ac.il/~vaidman/lvhp/m13.pdf

@J O Linton Noting where the photon landed on the screen amounts to what Aharonov and Vaidman call a postselection. Similarly, they identify preparation with a preselection.

Normally a description of a quantum system involves only a preselection state. In the paper above, they present a description of a quantum system involving both a postselection and preselection state, and identify intermediate values that are indeed not bound by the HUP. These values, however, cannot be resolved by generic measurements.

[edit] - Unfortunately the link seems to have died. Here is the (paywalled) link https://journals.aps.org/pra/abstract/10.1103/PhysRevA.41.11

Here is an accessible review of this line of thinking https://arxiv.org/pdf/quant-ph/0105101

 

[Lord Jestocost]

TL;DR Summary: Does the fact that you can measure the lateral momentum of a photon passing through a single slit contradict the HUP?

In their articles on the Heisenberg Uncertainty Principle Wikipedia says "there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known"; Britannica says " the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory. The very concepts of exact position and exact velocity together, in fact, have no meaning in nature."

In a previous thread Single slit diffraction I suggested that the unpredictability in the trajectory of a single photon passing through a single slit could be explained by saying that the slit imposes limits on the position of the photon in the Y direction thereby introducing an uncertainty in the Y momentum causing the diffraction. Granted this, when the photon lands on the screen and its deviation is noted, the lateral momentum can easily be calculated. We now know both the position and momentum of the photon as it passed through the slit to an accuracy much greater than that allowed by the HUP. How is this consistent with the statements quoted above?

Maybe, the following might be of help for an undertanding. Werner Heisenberg in his 1930 book "The Physical Principles of the Quantum Theory" (p. 20):

“The uncertainty principle refers to the degree of indeterminateness in the possible present knowledge of the simultaneous values of various quantities with which the quantum theory deals; it does not restrict, for example, the exactness of a position measurement alone or a velocity measurement alone. Thus suppose that the velocity of a free electron is precisely known, while the position is completely unknown. Then the principle states that every subsequent observation of the position will alter the momentum by an unknown and undeterminable amount such that after carrying out the experiment our knowledge of the electronic motion is restricted by the uncertainty relation. This may be expressed in concise and general terms by saying that every experiment destroys some of the knowledge of the system which was obtained by previous experiments. This formulation makes it clear that the uncertainty relation does not refer to the past; if the velocity of the electron is at first known and the position then exactly measured, the position for times previous to the measurement may be calculated. Then for these past times $\Delta p\Delta q$ is smaller than the usual limiting value, but this knowledge of the past is of a purely speculative character, since it can never (because of the unknown change in momentum caused by the position measurement) be used as an initial condition in any calculation of the future progress of the electron and thus cannot be subjected to experimental verification. It is a matter of personal belief whether such a calculation concerning the past history of the electron can be ascribed any physical reality or not." [Bold by LJ]

 

[Dale]

on an interpretation where the quantum state describes the state of an individual system, the HUP is about possible states of an individual system: the fact that there is no quantum state in which both position and momentum are precisely defined becomes a restriction on possible states of individual systems

But I think that the important point is that the HUP is a restriction on possible states, not possible measurements. You can have a very precise position measuring device and a very precise momentum measuring device, and you can apply both to the measurement of a state regardless of what HUP says

 

[PeterDonis]

I think that the important point is that the HUP is a restriction on possible states, not possible measurements.

It's both. Since the state makes predictions about measurement outcomes, a restriction on states is a restriction on measurement outcomes. The question of interpretation arises because different interpretations give different meanings to the state, over and above the predictions about measurement outcomes. But the state makes predictions about measurement outcomes in all interpretations; that part is just basic QM.

 

[Haborix]

It's both. Since the state makes predictions about measurement outcomes, a restriction on states is a restriction on measurement outcomes. The question of interpretation arises because different interpretations give different meanings to the state, over and above the predictions about measurement outcomes. But the state makes predictions about measurement outcomes in all interpretations; that part is just basic QM.

I think Dale is correct, especially in light of the context of the rest of his post. If you said, statistics of measurement outcomes, then I could agree with you.

 

[PeterDonis]

If you said, statistics of measurement outcomes, then I could agree with you.

The state does predict statistics, but it also predicts, for example, that it is impossible to have a measurement outcome which gives a precise value for both position and momentum. Any such outcome would have to be associated with an eigenstate of the measurement operator with those properties, and there is no such state.

 

[PeterDonis]

You can have a very precise position measuring device and a very precise momentum measuring device, and you can apply both to the measurement of a state

Can you? How? What operator would that correspond to? What would its eigenstates be?

 

[Vanadium 50]

Obviously, one operator can't return two numbers.

But I am with Dale here. You can measure two non-commuting variables as well as you'd like. The issue is whether an identically prepared system will give you these same two numbers next time. And they won't.

 

[PeterDonis]

You can measure two non-commuting variables as well as you'd like.

Not in the same measurement on the same single system.

 

[Vanadium 50]

Again, "same measurement" runs into the same issues. One operator can't return two numbers. That's truth classically.

 

[PeterDonis]

"same measurement" runs into the same issues. One operator can't return two numbers.

It's not a matter of having one operator return two numbers. It's a matter of there not being any single operator in the first place that equates to "measure position with unlimited accuracy and momentum with unlimited accuracy" (or the equivalent for any other pair of non-commuting observables). You can prepare a bunch of systems identically, and measure the position of half of them with unlimited accuracy, and the momentum of the other half with unlimited accuracy, and the results can help you to characterize the state being prepared as precisely as you like. But you can't take one single system and measure both position and momentum on it with unlimited accuracy in a single measurement. To me that counts as a restriction of the HUP on possible measurements. That is the point I have been making.

 

[J O Linton]

Maybe, the following might be of help for an undertanding. Werner Heisenberg in his 1930 book "The Physical Principles of the Quantum Theory" (p. 20):

“The uncertainty principle refers to the degree of indeterminateness in the possible present knowledge of the simultaneous values of various quantities with which the quantum theory deals; it does not restrict, for example, the exactness of a position measurement alone or a velocity measurement alone. Thus suppose that the velocity of a free electron is precisely known, while the position is completely unknown. Then the principle states that every subsequent observation of the position will alter the momentum by an unknown and undeterminable amount such that after carrying out the experiment our knowledge of the electronic motion is restricted by the uncertainty relation. This may be expressed in concise and general terms by saying that every experiment destroys some of the knowledge of the system which was obtained by previous experiments. This formulation makes it clear that the uncertainty relation does not refer to the past; if the velocity of the electron is at first known and the position then exactly measured, the position for times previous to the measurement may be calculated. Then for these past times $\Delta p\Delta q$ is smaller than the usual limiting value, but this knowledge of the past is of a purely speculative character, since it can never (because of the unknown change in momentum caused by the position measurement) be used as an initial condition in any calculation of the future progress of the electron and thus cannot be subjected to experimental verification. It is a matter of personal belief whether such a calculation concerning the past history of the electron can be ascribed any physical reality or not." [Bold by LJ]

I think this post best answers my question. Thank you.

 

[Morbert]

Again, "same measurement" runs into the same issues. One operator can't return two numbers. That's truth classically.

For two compatible observables A and B, we can write down a third observable C = AB with an eigenbasis $\{|\psi_{\mu\nu}\rangle\}$ such that $C|\Psi_{\mu\nu}\rangle = AB|\psi_{\mu\nu}\rangle = a_\mu b_\nu |\psi_{\mu\nu}\rangle = c_{\mu\nu}|\psi_{\mu\nu}\rangle$. One operator, one number returned, but giving certain information about the associated observables A and B. If two observables are incompatible, no such basis exists.

 

[Vanadium 50]

I was thinking about an example. A single more or less monoenergetic electron source sends electrons through a microscopic hole in a collimator. Electrons are then bent in a magnetic field and impact on a high-resolution screen where they are detected.

This device determines x (constrained from the collimator position) and p (from the radius of the magnetic bend) when the electron exits the collimator. What it won't do, however, is give the same values for p for consecutive identically prepared electrons.

 

[PeterDonis]

This device determines x (constrained from the collimator position) and p (from the radius of the magnetic bend) when the electron exits the collimator.

Electron diffraction at the collimator when the opening is made narrow enough will prevent any violation of the HUP.

What it won't do, however, is give the same values for p for consecutive identically prepared electrons.

Yes, agreed.

 

[PeterDonis]

This device determines x (constrained from the collimator position) and p (from the radius of the magnetic bend) when the electron exits the collimator.

What state is attributed to the electron when it exits the collimator based on this data? Suppose it is a Gaussian. Then it is impossible to construct a Gaussian that has an x-spread and a p-spread which, combined, violate the HUP. The x Gaussian and the p Gaussian are Fourier transforms of each other; you can't pick the x-spread and the p-spread independently. The Fourier transform forces the two spreads to obey the HUP when combined. That is true for any state whatever. So no matter how you construct this experiment, you won't be able to get a result even for a single run on a single electron that violates the HUP: there simply doesn't exist any such state that you could attribute to the electron from any data you could acquire.