Conservation of momentum of single photon passing a slit

[zhanhai]

When a single plane-wave photon/electron passes a slit/orifice, its direction of travel becomes random. Although there is the well-known Uncertainty Principle, it is not a replacement of the law of conservation of momentum for the phonon/electron before and after passing the slit.

Question 1:
If the photon/electron having pasted the slit is measured as traveling along a direction different from that of the original plane wave, where does the momentum difference go? Does the difference go to the screen of the slit? (I totally have no idea on this.)

Question 2:

A deterministic understanding would be that the direction of the photon/electron after the slit is determined when the photon/electron passes the slit, albeit the determination of the direction is random. (So the wave of the photon/electron after the slit is a new plane wave pocket. )

A non-deterministic understanding would be that the direction of the photon/electron is NOT determined until the photon/electron is measured. (So the wave of the photon/electron after the slit is a spherical wave?)

Is the above description correct? And which of the two understandings is more "correct" (or accepted by the mainstream)?

 

[The_Duck]

Question 1:
If the photon/electron having pasted the slit is measured as traveling along a direction different from that of the original plane wave, where does the momentum difference go? Does the difference go to the screen of the slit? (I totally have no idea on this.)

Yes. If you want to think semiclassically, you can imagine that the reason the particle's transverse momentum can change because it has some probability of ricocheting off the side of the slit. This makes it obvious that the momentum change is transferred to the screen.

A deterministic understanding would be that the direction of the photon/electron after the slit is determined when the photon/electron passes the slit, albeit the determination of the direction is random. (So the wave of the photon/electron after the slit is a new plane wave pocket. )

A non-deterministic understanding would be that the direction of the photon/electron is NOT determined until the photon/electron is measured. (So the wave of the photon/electron after the slit is a spherical wave?)

Is the above description correct? And which of the two understandings is more "correct" (or accepted by the mainstream)?

The second understanding is more mainstream: the wave function spreads out in a sort of spherical wave. But the de Broglie-Bohm interpretation goes with the first, deterministic understanding. In that interpretation, there is still a spherical "pilot wave," but the particle exists at a definite, deterministically changing position within that wave.

 

[zhanhai]

Thank you The_Duck for the reply.

I just got an interpretation. The momentum of the photon does not change before and after the slit. The wave before the slit is plane wave; the wave after the slit is like an inflating halfsphere whose total momentum (vector integral) is the same as that of the photon before the slit. So wherever the photon is detected after the slit, the entire halfspere collapses on the detector so the momentum received by the detector is the total momentum of the halfspere, which is the same as the momentum of the photon before the slit.

Experiment like ARPES might be able to check the validity of such interpretation.

 

[The_Duck]

Maybe I don't understand you fully, but I think this interpretation is not right. The momentum of the photon does change as it passes through the slit. More precisely, the only waves with definite momentum are plane waves. So the incident plane wave has a definite momentum. A spherical wave can be written as a sum--that is, a superposition--of a bunch (an infinite number) of plane waves. So after passing through the slit, the wave does not have a definite momentum but is in a superposition of a bunch of different momentum states.

After the particle passes through the slit, the screen is similarly in a superposition of a bunch of different momentum states. The screen and the particle become entangled, such that if the particle is found to have, say, 5 units of transverse momentum, the screen will be found to have -5 units of transverse momentum, so that the total transverse momentum (which was originally zero) is conserved in each component of the superposition. When the particle is later detected, the whole system of particle+screen collapses into one of the states in this superposition. The particle *will* be found to have a different momentum than it had before passing through the slit (at least if it is not detected directly behind the slit). Similarly the screen will be found to have acquired the opposite of whatever extra momentum was gained by the particle.

(I am glossing over some details related to the fact that the screen must start out with some initial uncertainty in its momentum.)

 

[zhanhai]

Maybe I don't understand you fully, but I think this interpretation is not right. The momentum of the photon does change as it passes through the slit. More precisely, the only waves with definite momentum are plane waves. So the incident plane wave has a definite momentum. A spherical wave can be written as a sum--that is, a superposition--of a bunch (an infinite number) of plane waves. So after passing through the slit, the wave does not have a definite momentum but is in a superposition of a bunch of different momentum states.

After the particle passes through the slit, the screen is similarly in a superposition of a bunch of different momentum states. The screen and the particle become entangled, such that if the particle is found to have, say, 5 units of transverse momentum, the screen will be found to have -5 units of transverse momentum, so that the total transverse momentum (which was originally zero) is conserved in each component of the superposition. When the particle is later detected, the whole system of particle+screen collapses into one of the states in this superposition. The particle *will* be found to have a different momentum than it had before passing through the slit (at least if it is not detected directly behind the slit). Similarly the screen will be found to have acquired the opposite of whatever extra momentum was gained by the particle.

(I am glossing over some details related to the fact that the screen must start out with some initial uncertainty in its momentum.)

It seems that the non-deterministics is of different orders. That the momentum of the photon after the slit is uncertain is 1st order, and that the amount of momentum exchanged between the photon and the slit cannot be determined until the photon is detected is second (or higher) order. I agree with the 1st order but disagree with the second order.

My view is that interpretation based on entanglement is often risky. Even if entanglement does exist (it seems controversial even in this forum), the spatial range in which entanglement can be realized is still in question. Detector can be far away from the slit, far enough that the slit can be actually removed before the photon reachs the detector and is detected; thus, the role of entanglement would be in question.

My previous description of the momentum process is not accurate. It should be that:

the wave pocket after the slit is a superposition of plane waves (momentum states) like ∫G(k)cos(kr-ωt+θ(k))dk (where k and r are vectors), and the wave before the slit is a nearly plane wave packet centered at k0. When the photon hits a detector (atom or so), only interactions led to by those momentum state components with k≈k0 could be realized. Interactions led to by the remaining momentum states, with k≠k0, could not be realized, ALTHOUGH they have non-zero matrix elements, because the law of momentum conservation is not satisfied.