Heisenberg imcertainty principle (get it)

[Antiphon]

Dan, I had to chop major sections of the quotation for byte limitation
reasons. I mostly don't quote those sections where I agree with you.
Also, I'm rushed and the proffreading is sloppy. Sorry.

This sounds too deterministic to me. How do you conserve that which you do not have? If the particle does not have a definite momentum between the slit and the screen, how do we conclude that a second position measurement retroactively puts the particle into a definite momentum state?

Once the particle lands and is detected, it there is a net momentum which
we can discuss as if it had had a classical history. This is what ZapperZ
has been insisting and he is right, but only after the particle lands. We've
gotten into one of the subtler areas of QM which I have seen described as
a correspondence principle, though not the idea that as a system becomes
larger it will transtition to classical behavior. Rather, this statement of
correspondence is that the notion of classical ideas like position and
momentum are not obsolete but their realms of applicability are restricted
by exactly the HUP.

I am not disputing the idea of entanglement. If we really did manage to precisely measure the momentum of the particle, then I suppose the slit should exhibit a recoil to conserve system momentum, if the state of the slit had remained entangled with the particle. But the slit is such a complex system, with all kinds of interactions going on while the particle is traversing the gap that I suspect entanglement would have to be considered in the other direction. The slit is the thing that is going to interact first with something else and that interaction is likely to force it, and therefore the particle, into a definite state.

When setting up a one-particle diffraction problem, the slit is usually
treated as a boundary condition, not a true quantum system with it's own
degrees of freedom. We could make a slit-like diffraction scenario out of
two additional particluate scaterers but the essence of (that) question
is the same- Which way do the dscattering particles recoil if momentum
is to be conserved? I believe it only makes sense to say they don't
recoil in a particlular way if the interacting particle hasn't got a particular
momentum.

I don't really want to focus on this unless it is absolutely necessary to understand what happens to the particle. But it does make me wonder if the particle can be forced into a definite state by entanglement because of interactions of the slit with the rest of the universe. Instead I want to pursue the evolution of the state of the particle without ever again worrying about the momentum of the slit.

Ok. I think the answer to the first part of you statement (question) here is
that if the particle is forced into a particular state, then you have made
a measurement of sorts, and not a diffraction of the sort we're contemplating.

I understand why ZapperZ is uncomfortable with the idea that a position measurement cannot be used to infer momentum. I am not arguing that it is impossible to use position to discriminate momentum. I am arguing that the single slit experiment is not consistent with spatially based momentum discrimination. I also think single slit diffraction has nothing to do with the ARPES experiments, except perhaps as an ultimate limit on resolution that has not even been approached.

You're right. Where you are missing it as regards Zapper's situation is
that he was never measuring momentum in a quantum sense. He only inferred it after-the-fact , a situation which must be consistent
with a classical description.

I'm not sure I am going to get this exactly right on the first try, even if I am on the right track, but I'm going to throw it out there for discussion. The question I want to pursue is, under what conditions can position measurements be used to deduce momentum?

My view of this is that the "classical" momentum measurement is the one
Zz is making- two position measurments at known times. When you do
this, even on quantum systems, you still get a classically consistent picture
for the final outcome which is this correspondence I referred to. This is not
in conflict with the HUP because that only refers to the uncertainty prior to
the second measurment.

In the single slit scenario we are all assuming that before the slit confinement the particle is in a highly localized momentum state. We all still believe there is a connection between momentum and direction of motion, both classically and in QM. We agree that when the particle leaves the slit its momentum is spread because the slit has imparted unknown momentum to the particle. I would say that since QM says this momentum is not specific, but can be cast as a superposition of momentum eigenstates, that conservation of momentum demands that each of these components be conserved. In other words, whatever linear combination of momentum eigenstates was prepared when the particle passed through the slit, it will have that same linear combination at all points in space and time between the slit and the screen.

I agree. I describe this rather imprecisely as the slit having an indefinite
recoil. What I really mean is that the slit by becoming entangled with
the particle acquires the same superposition, and there is a one-for-one
momentum conserving state of the slit with those of the particle.

Nevertheless, once the particle hits a screen, the slit must "decohere" or
acquire a "consisten history" or however you want to describe it- but the
other momentum eigenvalues cease to exist as possibilites any longer.

The superposition of momentum eigenstates can be expressed as a wave packet that spreads spatially with time while conserving the individual momentum components. Any spatial detector (screen) that I put in front of this wave packet is going to detect a hit at some location with the probability of a hit at anyone location determined by the spatial probability density function of the particle. My question is, does this detection by the screen tell me that the particle followed a trajectory along a line from the slit to the screen, implying that the particle had (or acquires retroactively) a particular momentum all the time? It sounds to me like you are saying it did, but that the path and its associated momentum did not exist until the second localization took place.

I think my above paragraph answers this somewhat. The particle did not
follow a classical trajectory but the outcome at the end can be analyzed
(as regards conservation laws) as if it did . As I said, the slits
states became a blur to match those of the particle in a conservation
sense.

I have a hard time with that idea. If the particle had localized momentum between the slit and the screen it would not have been spatially localized to a linear path between them. If it were in a transverse momentum eigenstate, it would have had an infinitely broad transverse spatial probability distribution, which means I could have detected it hitting at any location on the screen. I don't think it is possible to conclude that the momentum vector of the particle points from the slit to the location of the screen detection based on two position measurements. This is why in an earlier post I questioned whether a momentum measurement had been made at all.

The particle could not have had localized momentum as you pont out.

It was the second measurment which "determined" the final position of the
particle. This position then implies a classical momentum. The
particle didn't have it in flight, but once it hit the screen, the die was
cast. The screen has to have a matching state or globally momentum conservation is violated. I'm sottry to bring up the entanglement again
but that's the only way it makes sense to me. And if you think about it,
this situations is just like the EPR setup except we are not worried about
the causality but the consistency of conserved quatities at both ends of
the entangled system.

Now the slit comes into the picture... There is nothing in QM that tells us how a particle makes its way form one position allowed by its wave function to another position.

I agree completely.

I cannot do that in this single slit experiment with a position detector. If I have prepared a particle in such a way that it is in a state of superposition of [...]
First, if you want to explore the physics of the process of generating photoelectrons from a sample, this article is accessible online and goes into some detail about how you know the momentum components of photoelectrons transverse to a sample surface.

Unless I missed a point of disagreement, I believe I agree with everything
you say above Dan.

I make no pretense of having grasped that in detail, but I understand it sufficiently well to recognize that most of the calculations are based on the assumption that the photoelectrons are being emitted from the sample in momentum eigenstates. From a QM perspective that would imply infinite spatial non-locality. Of course the authors are implying no such thing. What they are really saying, in my interpretation, is that emanating from a small volume in the sample that is excited by incident photons is an electron that can be characterized by a wave function that corresponds to a wave packet of nearly constant momentum and sufficient spatial limitation so that as wave packets with different central momentum migrate away from the spot they will become spatially separated much the same way as white light would be separated by a grating. Figure 8 and the associated text gives a pretty good description of the physical arrangement. In particular I wanted to find out something about the nature and dimensions of a typical detector. That can be find here

http://www.gammadata.se/ULProductFiles/Scienta_R4000_1.pdf

The parameters of particular interest are: The unique 0.1 mm wide slit
offers possibility of measuring extremely high energy resolution. and the typical electron energies in the range of 1 to 100 eV in their representative graphs for the angle resolution mode. The deBroglie wavelength of a 1eV electron is about 12nm, so we are talking about a slit that is on the order of 10,000 electron wavelengths or more. The other dimension of the slit is the one that would create momentum spreading in the direction of momentum resolution, and it is even wider, though not specified. In other words, single slit diffraction is not an issue. Once those electrons get out of the sample they are headed off into space with highly localized momentum that permits spatially separated detections to be associated with nearly unique momentum values.

Agreed...

So, at least for the moment, I am completely comfortable with ZapperZ believing that he is discriminating momentum in his experiments and finding useful information about the energy/momentum relationships in the materials he is studying, and I am comfortable that ARPES has nothing to do with the single slit experiment and its relationship to HUP.

Ok. You now know far more about ARPES than I do and I take your word for it.

I am also still confident that in the single slit scenario a hit on the screen does not constitute a momentum measurement. My comfort may only last until the next message is posted, but so be it.

Ok, here's where we part company slightly I think. It's not a momentum
measurmentat the quantum level. As you pointed out, the measurement
had nothing to do with determining the angle of arrival of the wave packet
because it was in fact a position measurement.

And I agree with you, if you ignore the slit and make a position measurement
you know absolutely nothing about the momentum regardless of the HUP.

What I assert here is that the momentum which is inferred by making two
quantum position measurements is a valid (essentially classical) momentum measurement. The
fact that the particle did not have a specific trajectory or momentum
should not deter us from treating the system's result as classical after the
fact.

 

[Antiphon]

Do you think it is possible that what constitutes a "momentum measurement" could depend on the way one interprets QM? Or do you both discount that possibility?

I rather think we're making very thin slices off of the sausage called
"how to interpret the relationship betwen classical and quantum mechanics,"
and the way in which the quantum and the classical are made consistent
with one another where it is necessary to do so. But everyone in this
thread has a very good handle on the basics.