Heisenberg imcertainty principle (get it)

[ZapperZ]

I truly do not understand what the problem here is.

I have ALREADY explained what I meant by "simultaneous", so I don't understand why you still bringing it up. It ISN'T what you accused me of doing.

Secondly, and this is even more puzzling because the practice is very rampant. How do you think we "detect" the properties of electrons, or other particles in the first place? More often than not, we detect them by observing where they are! We do this in SEM, STM, etc. I measure the energy of an electron by how much it bends in a magnetic field, and then I look at WHERE it lands on a detector! This tells me how much it has bent! Yet, from the way you are tell me here is that this is NOT what its energy is as a free, plane-wave particle, that between the moment it enters the magnetic field till it is detected, its momentum and energy are still in a superposition of values and so what is being detected is some "detection values".

I have repeated this many times, that I make no assumption of what happened between the slit and the detector. All I'm saying is that THAT electron that hit the detector has THAT momentum when it hits the detector. If by looking at the image on the detector and deducing the momentum is WRONG, then we have been wrong in MANY, MANY other techniques and detection schemes, especially in high energy physics because they make even MORE strong assumptions about the trajectory of the particle from the collision point to the detectors.

Zz.

 

[Gokul43201]

For those with a little bit of linear algebra under their belts :

Let A, B be a pair of non-commuting operators. What this means is that A and B do not posses the same eigenfunctions (eigenkets). We first define the operator \Delta A \equiv A - \langle A \rangle I. Where the expectation value of A with respect to some state |a \rangle, is defined as \langle A \rangle = \langle a | A | a \rangle. This number tells you what A will be measured as, on average, over several repeated measurements performed on the system, when prepared identically.

Now, we define an important quantity - the variance or mean square deviation, which is \langle ( \Delta A ) ^2 \rangle. This quantity is no different from the variance in any statistical collection of data. Plugging in from above :

\langle ( \Delta A ) ^2 \rangle = \langle (A - \langle A \rangle I)^2 \rangle = \langle A^2 \rangle - \langle A \rangle ^2 ~~~-~(1)

Let |x \rangle be any arbitrary (but normalized) state ket. Let :

|a \rangle = \Delta A ~ |x \rangle
|b \rangle = \Delta B ~ |x \rangle

First we apply the Schwarz inequality (which is essentially a result that is two steps removed from saying that the length of a vector is a positive, real number) : \langle a |a \rangle \langle b |b \rangle \geq | \langle a |b \rangle |^2 to the above kets (keeping in mind that \Delta A~, ~\Delta B are Hermitian), giving :

\langle (\Delta A)^2 \rangle \langle (\Delta B)^2 \rangle \geq |\langle \Delta A \Delta B \rangle | ^2 ~~~-~(2)

Next we write

\Delta A \Delta B = \frac{1}{2}(\Delta A \Delta B - \Delta B \Delta A) + \frac{1}{2}(\Delta A \Delta B + \Delta B \Delta A) = \frac{1}{2}[\Delta A, \Delta B] + \frac{1}{2}\{ \Delta A, \Delta B \} ~~~-~(3)

Now, the commutator

[\Delta A,~ \Delta B] = [A - \langle A \rangle I,~B - \langle B \rangle I] = [A,B] ~~~-~(4)

And notice that [A,B] is anti-Hermitian, giving it a purely imaginary expectation value. On the other hand, the anti-commutator \{ \Delta A,~ \Delta B \} is clearly Hermitian, and so, has a real expectation. Thus :

\langle \Delta A \Delta B \rangle = \frac{1}{2}\langle [A,B] \rangle + \frac{1}{2} \langle \{ \Delta A,~ \Delta B \} \rangle ~~~-~(5)

Since the terms on the RHS are merely the real and imaginary parts of the expectation on the LHS, we have

| \langle \Delta A \Delta B \rangle |^2 = \frac{1}{4}| \langle [ A,B] \rangle |^2 + \frac{1}{4} | \langle \{ \Delta A,~ \Delta B} \rangle |^2 \geq \frac{1}{4}| \langle [A,B] \rangle |^2~~~-~(6)

Using the result of (6) in (2) gives :

\langle (\Delta A)^2 \rangle \langle (\Delta B)^2 \rangle \geq \frac{1}{4} |\langle [A,B] \rangle | ^2 ~~~-~(7)

The above equation (7), is the general statement of the Heisenberg Uncertainty Principle. So far, it is nothing more than the statement of a particular property of certain specifically constructed hermitian matrices - namely, those of the form\Delta H, constructed as above.

Notice that if the operators A, B comute (ie: [A,B] = 0), then the product of the variances vanish, and there is no uncertanity in measuring their observables simultaneously. It is only in the case of non-commuting (or incompatible) operators, that you see the more popular form of the Uncertainty Principle - the one where the product of the variances does not vanish. Specifically, in the case where A = \hat{x_i}~,~~B = \hat{p_i}, we have

[\hat{x_i},\hat{p_i}] = i \hbar ~~~-~(8)

This result follows from essentially two observations :

(i) The infinitesimal translation operator, \tau (d \mathbf{x}), defined by \tau (d \mathbf{x}) |\mathbf{x} \rangle \equiv |\mathbf{x} + d \mathbf{x} \rangle can be written as

\tau (d \mathbf{x}) = I - i\mathbf{K} \cdot d \mathbf{x}

(ii) K is an operator with dimension length ^-1, and hence, can be written as \mathbf{K} = \mathbf{p} / [action]. The choice of this universal constant with dimensions of action (energy*time) comes from the de broglie observation k = p/ \hbar. So, writing \tau (d \mathbf{x}) = I - i\mathbf{p} \cdot d \mathbf{x} /\hbar leads to the expected commutation relation , [\hat{x_i}, \hat{p_i} ] = i \hbar. Plugging this into (7) gives the correct version of the popular form of the HUP :

\langle (\Delta x_i)^2 \rangle \langle (\Delta p_i)^2 \rangle \geq \frac{1}{4} \hbar ^2 ~~~-~(HUP)

Illegally taking square roots above and forgetting that we are talking about variances and expectations, is what leads to popular misconceptions about intrinsic uncertainties in single measurements.

 

[Gokul43201]

Another common misconception is that the HUP for position-momentum is merely a restatement of the x,p commutator.

No, ~ \Delta x \Delta p \neq [x,p].

 

[Igor_S]

...
We first define the operator \Delta A \equiv A - \langle A \rangle.
...

Just a note to avoid (possible) unnecessary confusion: \Delta A \equiv A - \langle A \rangle I, where I is identity operator (you have been perfectly clear in your derivation everywhere else). I know this is not usually explicitly written, but it cannot hurt to say it.

 

[Gokul43201]

Thanks, Igor. I'll make the changes to ensure that operators do not look like scalars anywhere.

 

[OlderDan]

Illegally taking square roots above and forgetting that we are talking about variances and expectations, is what leads to popular misconceptions about intrinsic uncertainties in single measurements.

Would you care to elaborate on that? I don't think there is any real problem about what is going to happen if repeated measurements are made under the same conditions. Repeated trials are going to result in a pattern of hits on the screen. The issue is focused on a single particle passing through a single slit being detected one time on a screen. You know the particle made it through the slit, and you know, to a precision determined by the detector, where the particle hit the screen. What do you know about the particle? What have you measured?

 

[Antiphon]

Illegally taking square roots above and forgetting that we are talking about variances and expectations, is what leads to popular misconceptions about intrinsic uncertainties in single measurements.

The more precisely the position is determined, the less precisely the momentum
is known in this instant, and vice versa.

Also, it would seem the two of you have a disagreement.

 

[Gokul43201]

OlderDan :

The Uncertainty Principle says nothing about what happens in a single measurement. What does one mean by the error or imprecision or uncertainty of a single data point ? It is meaningless. Even for a somewhat crude (although possibly illustrative) thought experiment one considers two data points at the extrema of the detection window and relates the spread between these data points to the spread between calculated values of the conjugate variable.

See the Gamma Ray Microscope thought experiment that Heisenberg came up with. When Heisenberg first proposed the thought experiment, he got it wrong, and had to be corrected by Bohr.

Antiphon : I shall not respond to claims based upon absolutely referenceless quotes. You'll have to do better than that.

 

[Gokul43201]

The Gamma Ray Microscope

http://www.aip.org/history/heisenberg/p08b.htm

 

[poolwin2001]

Well according to this thread all the problems I have done in high school are stupid.Like this in this thread:
https://www.physicsforums.com/showthread.php?t=83213 :

According to HUP \Delta x.\Delta p \geqq \frac{h}{4 \pi}. Pluging in values where \Delta x will be the size of the nucleus, we get
\Delta v greater than c ! We should not have velocities above c while here even the uncertainity in velocity is greater than c which indicates that our \Delta x is incorrect ==> e- can't be confined to the nucleus.
As the neutrons and protons differ in mass by about 10e3 the \Delta v for n/p doesn't come above c ! So they may exist inside the nucleus.

If this is the case,Why isn't HUP denoted as
\sigma_x.\sigma_p \geqq some k or something ?
where \sigma stands for standard deviation.

 

[OlderDan]

OlderDan :

The Uncertainty Principle says nothing about what happens in a single measurement. What does one mean by the error or imprecision or uncertainty of a single data point? It is meaningless. Even for a somewhat crude (although possibly illustrative) thought experiment one considers two data points at the extrema of the detection window and relates the spread between these data points to the spread between calculated values of the conjugate variable.

See the Gamma Ray Microscope thought experiment that Heisenberg came up with. When Heisenberg first proposed the thought experiment, he got it wrong, and had to be corrected by Bohr.

Antiphon : I shall not respond to claims based upon absolutely referenceless quotes. You'll have to do better than that.

This is not an answer to my question. The only reference I made to measurement in my previous post was in asking the question about what is being measured given one particle directed toward one slit, and one detection of the particle striking the screen on the other side of the slit.

Nevertheless, the "uncertainty" of a single measurement is not a meaningless concept. It means the same thing here that it means when you get out your ruler to find out how long to cut a board. You measure it once, and write down a number or mark the board, and you know that the measurement lacks precision because the instrument you used and your ability to read it are imperfect. You may not know exactly how to quantify the uncertainty, but if it's a good ruler and your eyes are better than mine you can be pretty confident that the measurement is accurate to far better than an inch, and probably not much better than 1/32. If you are good at using the saw to actually cut the board to the length you measured, you can make a living as a carpenter. If you are a precision machinist, you use better tools and get more precise results. You don't have to take a statistical average of many measurements to have a pretty good idea of the potential error of your one measurement.

In the problem at hand, the particle passed through a slit that is presumed to have a known width to arbitrary precision. Given our belief that in order to arrive at the screen the particle must have passed through the slit, we conclude that the particle was localized when it passed through the slit. We don't claim specific knowledge of where the particle passed through the slit, just that it made its way through the gap somewhere. We must accept that we have not located the position of the particle with greater precision than one slit width, but we can be confident that the slit width is a measure of the potential uncertainty in the position of the particle when passing through the slit.

When the particle hits the screen it leaves a mark, or excites one or a few cells of an array of detectors. The size of the cell apertures and/or collection of excited cells in the array gives us the same sense of uncertainty in the location of the strike that the width of the slit gives us about where the particle came through the slit. We don't have to run the same particle through the apparatus again and again to take a statistical average of hits to have a sense of the precision of the location of one hit. We all know that in the scanario at issue repeating the measurement would only defeat the purpose.

I see no value in bringing yet another thought experiment into the discussion, especially given the last paragraph of the link you posted. The only connection between the microscope and the slit apparatus is the collection apertures, and that's covered in the previous paragraphs.

A couple of us have attempted to talk about the wave or state function of the particle as it pertains to what can be know about one particle, and the response has been as if we are talking in some foreign language. I will readily accept that I am out of touch with current thinking on these issues, but I have read a lot of stuff in the last couple of days about particle tracks and quantum measurement theory and I have yet to find a single reference that dismisses the fundamental problem of limited knowledge of the "observables" of a single particle, and I have found repeated references to the HUP in connection with those limitations.

Nowhere have I said that it is impossible for any single measurement of one observable to yield a precise value. Nowhere have I implied that a particle passing through a single slit is going to be smeared across the detection screen when it gets there. What I have done is to make reference to the fact that a wave function that localizes a particle can be represented as a spectrum of momentum eigenfunctions, but most certainly not as a single momentum eigenfunction. Represent that in whatever equivalent way you want with non-commuting operators, state functions that are not simultaneously eigenstates of position and momentum- I don't care. The representation is not the issue.

The issue is, can I do something that will tell me with unlimited precision both the position of one particle and its momentum at the same moment in time? If the answer is yes, then I want to know where to find the supporting evidence, because I cannot find it. If the answer is no then I want the broadly accepted reference that dismisses that fundamental property of individual microscopic particles as unrelated to the HUP, because everything I have found so far keeps saying that it is.

 

[ZapperZ]

The issue is, can I do something that will tell me with unlimited precision both the position of one particle and its momentum at the same moment in time? If the answer is yes, then I want to know where to find the supporting evidence, because I cannot find it. If the answer is no then I want the broadly accepted reference that dismisses that fundamental property of individual microscopic particles as unrelated to the HUP, because everything I have found so far keeps saying that it is.

But OlderDan, is this really the issue that we started with? And is this really what *I* initially started with?

I mean, look at what is going on. A particle that initially only had a momentum in the z-direction moving towards the slit, after passing the slit, now has gained a y-component of the momentum, something it did NOT have before it went through the slit. The act of using the slit to determine its position has CHANGED the system in the sense that it introduced an added momentum component. This should not be a surprise to anyone.

I THEN measure this momentum. And as far as I can tell, it is THIS ability that you are disputing. It would be silly for me to insist that this is the SAME momentum of the single particle when it enters the slit, because I'm measuring the component of momentum that it didn't have before! And I've given you at least a couple of citations and a slew of experimental techniques in which the momentum, energy, and other characteristics of an electron are deduced using the knowledge of where on a detector that electron hits. In fact, the Valla et al. Science paper was one of the top 20 most cited paper in 2001 primarily due to the introduction of the energy and momentum distribution curve in a photoemission measurement.

BTW, your explanantion of the uncertainty in your "ruler" is identical to what I have said about the uncertainty in instrumentation. This isn't the HUP. The tick marks and ability to read your ruler doesn't change just because you make something else smaller or larger. This uncertainty is not what Gokul has derived.

Zz.

 

[ZapperZ]

Well according to this thread all the problems I have done in high school are stupid.Like this in this thread:
https://www.physicsforums.com/showthread.php?t=83213 :


If this is the case,Why isn't HUP denoted as
\sigma_x.\sigma_p \geqq some k or something ?
where \sigma stands for standard deviation.

If that is what you have "deduced" from this thread, then you have severely misread it.

Zz.

 

[Igor_S]

Secondly, and this is even more puzzling because the practice is very rampant. How do you think we "detect" the properties of electrons, or other particles in the first place? More often than not, we detect them by observing where they are! We do this in SEM, STM, etc. I measure the energy of an electron by how much it bends in a magnetic field, and then I look at WHERE it lands on a detector! This tells me how much it has bent! Yet, from the way you are tell me here is that this is NOT what its energy is as a free, plane-wave particle, that between the moment it enters the magnetic field till it is detected, its momentum and energy are still in a superposition of values and so what is being detected is some "detection values".

I have repeated this many times, that I make no assumption of what happened between the slit and the detector. All I'm saying is that THAT electron that hit the detector has THAT momentum when it hits the detector. If by looking at the image on the detector and deducing the momentum is WRONG, then we have been wrong in MANY, MANY other techniques and detection schemes, especially in high energy physics because they make even MORE strong assumptions about the trajectory of the particle from the collision point to the detectors.

I think the detection techniques are actually measuring classical quantities. When you measure how much particle has bent in a magnetic field, you use R = mv / eB (at least in a simple cyclotron). This is classical equation. I don't think HUP says anything involving classical momentum (p = mv). Classical momentum appears in QM only as expectation value of QM momentum. Proper way to determine QM momentum would be from interference pattern (like I think OlderDan said before), by measuring the distances between max or minumum intensity points. I'm not 100% sure, so correct me if I'm wrong.

But why can we then see particle track in detector, I am not sure, but maybe it's because a particles in this experiments have so much bigger momentum in certain direction ? (so you can actually approximate that other components of momentum are zero)

 

[ZapperZ]

I think the detection techniques are actually measuring classical quantities. When you measure how much particle has bent in a magnetic field, you use R = mv / eB (at least in a simple cyclotron). This is classical equation. I don't think HUP says anything involving classical momentum (p = mv). Classical momentum appears in QM only as expectation value of QM momentum. Proper way to determine QM momentum would be from interference pattern (like I think OlderDan said before), by measuring the distances between max or minumum intensity points. I'm not 100% sure, so correct me if I'm wrong.

But why can we then see particle track in detector, I am not sure, but maybe it's because a particles in this experiments have so much bigger momentum in certain direction ? (so you can actually approximate that other components of momentum are zero)

But ALL "measurements" are "classical". We are trying to determine classical properties such as "position", "momentum", "energy", etc. When we make a measurement, we force the system to interact with large degree of freedom that causes certain degree of decoherence. What I indicated is no different.

Electrons in solids such as metals have a superpostion of momentum/energy, etc. The principle of photoemission says that we CAN measure accurately the in-plane momentum and energy of the emitted photoelectrons, and that what we detect on our detector are those two values. And not only that, these two values represent the in-plane energy and momentum of that electron while it was in that material!

http://arxiv.org/abs/cond-mat/0209476
http://arxiv.org/abs/cond-mat/0208504

Zz.

 

[seratend]

I think the confusion comes from the interpretation of zapper and older Dan words (and sometimes in forgetting we live in a 3D/4D world).
A measurement by the slit, defines a position measurement (x,y) in the “transversal” direction of the moving particle (direction z). From this measurement result we deduce the momentum of the particle thanks to the preparation (e.g. energy conservation, all particles have the same energy: see note below).
We have 2 commuting observables: the transversal position (x,y) and the momentum pz of the wave packet => the Heisenberg principle keeps working (what zapper and older dan says are ok, provided we understand their mapping into the physical reality and QM formalism).
If we have a detector behind the slit, it will also do a z position measurement (on the preparation given by the slit plate and the particle state). And we can say, yes “before” the measurement result of this detector, the particle has a momentum pz defined by the position (x,y) of the slit (but not the position z) (hence a position (x,y) and momentum pz) and yes “after” the detector measurement (the click), the particle is located at (x,y,z) with an undefined momentum.

I hope, that my words have not added more confusion .

Note: we can infer the momentum pz from the transversal position measurement because the preparation of the particle separates spatially (transversal direction) with ~100% confidence –the wave packets of different momentums pz. This is why we put a plate with a pin hole before the double slit plate of the double slit experiment for example (we can say that the two plates measure or prepare the momentum of the particle).

Hoping my contribution may help in ansering the different questions,

Seratend.

EDIT: in the note above, different momentums pz, mean different z axes (sorry for the confusion). I hope one has assumed the correct meaning : )

 

[Hans de Vries]

Note: we can infer the momentum pz from the transversal position measurement because the preparation of the particle separates spatially (transversal direction) with ~100% confidence

This is the point of the confussion. It's because this is an average momentum.

From the orthodox Copenhagen interpretation one might argue that this is not
the same as the instantaneous momentum. In the line of: "a particle may have
a ΔE for a certain Δt", one might argue that it may have a fluctuating Δp which
averages out over a longer traject.


Regards, Hans

 

[Hans de Vries]

There is a common idea that it is impossible to determine the energy E and
momentum p of a particle from an infinitesimal small region of the wave
function because of the Fourier Transform relation.



However, If we write for a wave-packet at rest:

\ Q_{\{x\}}\ \ e^{ -iE_0t/\hbar}

Were Q is the shape of the wave function (e.g. the Gaussian) the we can
determine the value of E at each point in space time simply by differentiating
in time.


I we then handle this as a moving wave-packet via the Lorentz transform:

\ Q'_{\{x,t\}}\ \ e^{ipx/\hbar}\ \ e^{ - iEt/\hbar }, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{with: } Q'_{\{x,t\}} = Q \{ \gamma (x-vt) \}

Then we can infer E and p locally exact at each point in space and
time by differentiating in x and t.


Regards, Hans

 

[ZapperZ]

There is a common idea that it is impossible to determine the energy E and
momentum p of a particle from an infinitesimal small region of the wave
function because of the Fourier Transform relation.



However, If we write for a wave-packet at rest:

\ Q_{\{x\}}\ \ e^{ -iE_0t/\hbar}

Were Q is the shape of the wave function (e.g. the Gaussian) the we can
determine the value of E at each point in space time simply by differentiating
in time.


I we then handle this as a moving wave-packet via the Lorentz transform:

\ Q'_{\{x,t\}}\ \ e^{ipx/\hbar}\ \ e^{ - iEt/\hbar }, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{with: } Q'_{\{x,t\}} = Q \{ \gamma (x-vt) \}

Then we can infer E and p locally exact at each point in space and
time by differentiating in x and t.


Regards, Hans

On the other hand, E (actually H, the Hamiltonian) and p commute for a free particle. So it should be of no mystery that there are instances where they both can be known with the same uncertainty "simultaneously".

Zz.

 

[seratend]

This is the point of the confussion. It's because this is an average momentum.

From the orthodox Copenhagen interpretation one might argue that this is not
the same as the instantaneous momentum. In the line of: "a particle may have
a ΔE for a certain Δt", one might argue that it may have a fluctuating Δp which
averages out over a longer traject.

Regards, Hans

Yes, however I have to emphasize we are playing with words (a matter of interpretation): delta pz. delta z ~ hbar => having 2 plates with 2 slits with an infinite distance between plates (=> delta z --> +oO) allows one to have a delta pz as small as one wants (provided some technical feasibilities).
In the formal limit delta pz=0 => the mean value equals the value of the momentum, we cannot distinguish them.
(what zapper said: we can measure the momentum pz with the precision we want depending only on the technology feasibility).

Seratend.

 

[ZapperZ]

(what zapper said: we can measure the momentum pz with the precision we want depending only on the technology feasibility).

Seratend.

Actually, I said py, since in my example, the particles are moving in the z-direction when it hits the slit that is oriented along the x-direction. So the slit has a width Delta(y).

But, those are just details... :)

Zz.

 

[Antiphon]

I apologize for butting into the thread again but I think I spot the problem now.

But OlderDan, is this really the issue that we started with? And is this really what *I* initially started with?

I mean, look at what is going on. A particle that initially only had a momentum in the z-direction moving towards the slit, after passing the slit, now has gained a y-component of the momentum, something it did NOT have before it went through the slit.

This is incorrect. It's true that there was no y-compnent before. But after
the slit there is NOT a non-zero y-component. There are many possible y-components
(both + and - values (i.e. directions)). This is NOT something that can be
measured. Your measurement device will return a number, not a quantum
superposition of possible numbers. The statistical distribution of the
superposition will be revealed by measuring many particles. But the
uncertainty of the (clasical and in-fact non-existent) momentum value is inherent
in each individual trial.

The act of using the slit to determine its position has CHANGED the system in the sense that it introduced an added momentum component. This should not be a surprise to anyone.

This is not right. A new momentum component was not added. The
wavefunction which had been composed of one wavlength (corresponding
to a specific value of the classical momentum) has been changed into a
superposition of many possible momenta. If this were right, it would be
equivalent to saying that the particle has a well-define classical trajectory
from the slit to the detector. It doesn't.

I THEN measure this momentum. And as far as I can tell, it is THIS ability that you are disputing..

No. You can perform this measurement. It's just that the momentum you
measure was NOT the momentum of the particle on it's way to the detector.
Before this measurement, that particle did not posess that or any other
value for its momentum. If it did, it would have had a classical trajectory.

 

[Igor_S]

I forgot to mention in my earlier post:

"Among other things, the wave character of matter [i.e. that in QM particles are guided by the field \psi (x,t)] manifests itself in the fact that there is a direct connection between position and momentum determination in microscopic physics, namely, we are not able to measure the exact position and momentum of particle, simultaneously. The amount of uncertainty is given by HUP."
- W. Greiner, "Quantum mechanics, an introduction" (4ed., Springer, 2001)

This is cleary the opposite what Zapper said (in some of the previous posts). However:

"Please understand what the uncertainty principle means: Like position mea-
surements, momentum measurements yield precise answers--the "spread"* here refers to the fact that measurements on identical systems do not yield consistent results. You can, if you want, prepare a system such that repeated position measurements will be very close together (by making \psi a localized "spike"), but you will pay a price: Momentum measurements on this state will be widely scattered. Or you can prepare a system with a reproducible momentum (by making \psi a long sinusoidal wave), but in
that case position measurements will be widely scattered. And, of course, if you're in a really bad mood you can prepare a system in which neither position nor momentum is well defined..."
- D. Griffiths, "Introduction to quantum mechanics" (PH, 1995)

But ALL "measurements" are "classical". We are trying to determine classical properties such as "position", "momentum", "energy", etc. When we make a measurement, we force the system to interact with large degree of freedom that causes certain degree of decoherence. What I indicated is no different.

Hmm... the proper way to detemine momentum (given by p = h / \lambda) is to take many particles hitting the screen and measure distance between maximum and minimum. And that's where I see I cannot determine position at that same screen because in order to measure momentum, I had to take many particles. And that's why HUP can experimentally be seen only with a bunch of measurements (not with only one). BUT, I can say that particle does not have well-defined momentum between the slit and the screen (it's in a superposition; whatever that physically is). In order to check something happened with momentum, I have to make many measurements. How else am I going to this prove to someone ?

I think you have made a mistake (or I figured you out wrong) when you said that CCD screen at the same time measures both position and momentum. That may not be proven by using HUP, but using basic postulates of QM (projection postulate). What I can see from 1 single measurement is something like "projection" into some state (I'm not sure what word to use). I can definitely say "something" has happened to momentum, because the value I will most probably get will be different from original one, or it has changed, but what really happened? Will I always get that new value given same initial conditions? I cannot answer that question with only 1 measurement "under my belt". But, and I think we can agree here, we can say "momentum observed is randomly distributed" only when we made enough measurements under identical conditions.


---
* he refers to \sqrt{ < \Delta x ^2 >}

 

[ZapperZ]

"Among other things, the wave character of matter [i.e. that in QM particles are guided by the field \psi (x,t)] manifests itself in the fact that there is a direct connection between position and momentum determination in microscopic physics, namely, we are not able to measure the exact position and momentum of particle, simultaneously. The amount of uncertainty is given by HUP."
- W. Greiner, "Quantum mechanics, an introduction" (4ed., Springer, 2001)

This is cleary the opposite what Zapper said (in some of the previous posts). However:

"Please understand what the uncertainty principle means: Like position mea-
surements, momentum measurements yield precise answers--the "spread"* here refers to the fact that measurements on identical systems do not yield consistent results. You can, if you want, prepare a system such that repeated position measurements will be very close together (by making \psi a localized "spike"), but you will pay a price: Momentum measurements on this state will be widely scattered. Or you can prepare a system with a reproducible momentum (by making \psi a long sinusoidal wave), but in
that case position measurements will be widely scattered. And, of course, if you're in a really bad mood you can prepare a system in which neither position nor momentum is well defined..."
- D. Griffiths, "Introduction to quantum mechanics" (PH, 1995)

And what Griffiths said is exactly what I said in my journal entry. The spread in where the many dots one sees at the detector will get larger as one makes the slit smaller. This is the HUP at work. I don't think this is in disput by anyone here. However, you cannot see this HUP spread IF you only make ONE dot. The "spread" of this dot (i.e. the accuracy of my measurement of were it hits the detector) appears to be in dispute when I say that it is NOT involved in the HUP and that this location also tells me the momentum of this particle.

Hmm... the proper way to detemine momentum (given by p = h / \lambda) is to take many particles hitting the screen and measure distance between maximum and minimum. And that's where I see I cannot determine position at that same screen because in order to measure momentum, I had to take many particles. And that's why HUP can experimentally be seen only with a bunch of measurements (not with only one). BUT, I can say that particle does not have well-defined momentum between the slit and the screen (it's in a superposition; whatever that physically is). In order to check something happened with momentum, I have to make many measurements. How else am I going to this prove to someone ?

And if you make a measurement, you get just ONE momentum, not a superposition of momentum, for that one particle. You shoot another particle under the identical condition, you get ANOTHER different momentum value, etc...etc. That fact that you do get ALL these different momentum (but NOT all at once from ONE particle) is the manifestation of the superposition of various momentum. But you don't get all of these from measuring just ONE particle.

I think you have made a mistake (or I figured you out wrong) when you said that CCD screen at the same time measures both position and momentum.

Whoa! Hang on! You read it wrong! I said that the "position measurer" is the slit. The position on the DETECTOR/SCREEN corresponds to the "momentum measurer". I deduce the momentum of the particle from where it hits the screen. I do not use it to measure the ORIGINAL position, i.e. this is not the "x" in [x,p].

Look at the Photoemission references I gave you. When you see images in there that is similar to my avatar, note that these are RAW CCD images made by electrons hitting the "screen". The ONLY thing done to them is to calibrate the location of where they hit the screen with energy and momentum of each of those particles. The location on where the hit the detector is not the "position" operator.

Zz.

 

[ZapperZ]

This is incorrect. It's true that there was no y-compnent before. But afterthe slit there is NOT a non-zero y-component. There are many possible y-components(both + and - values (i.e. directions)). This is NOT something that can be
measured. Your measurement device will return a number, not a quantum
superposition of possible numbers. The statistical distribution of the
superposition will be revealed by measuring many particles. But the
uncertainty of the (clasically non-existenet) momentum value is inherent
in each individual trial.

Er.. Hello? THAT is the whole point! I'm measuring the system AFTER, AFTER, AFTER, AFTER it passes the slit!

Short of writing that in both CAPS and bold, I have no idea how else can I emphasize it. I'm NOT measuring the system BEFORE it enters the damn slit!

ISSUE AT HAND: You make a measurement of A. People say that AFTER, AFTER, AFTER, AFTER you do that, B is "smeared" and so undefined. That is only correct if one talks about our ability to PREDICT what B measurement is going to be. But this has been bastardized by many by saying that ONE single measurement by B AFTER, AFTER, AFTER, AFTER A is made will yield a SMEARING of value! This is horribly wrong! You only get a smearing of values upon REPEATED measurement of B becuase B will produce many DIFFERENT values for all those repeated measurement!

Are we finally getting this CLEAR? I don't mind defending what I said. I just don't care defending what I DIDN'T say.

Zz.

I think I've just popped a vein!

 

[Antiphon]

I'm NOT measuring the system BEFORE it enters the damn slit!

I didn't say you did.

I'm saying you claim a well-defined momentum exists between slit and
detector. It simply doesnt.

Edit: Maybe claim is too strong- you're implying it.
Edit 2: Above: read "well-defined" as a single numerical classical quantity.

 

[ZapperZ]

I didn't say you did.

I'm saying you claim a well-defined momentum exists between slit and
detector. It simply doesnt.

Then ALL ARPES results are crap! How do you explain that they explained the physics of material so well?

Zz.

 

[Antiphon]

Then ALL ARPES results are crap! How do you explain that they explained the physics of material so well?

Zz.

Sorry, I'm not familier with ARPES. But it's results are no doubt
valid because:

When the particle finally hits the detector and you detemine that it had
such-and-such y-compnent of the momentum, that's a correct summation
of the OUTCOME of the entire experiment.

That is, its not untrue to say
1) particle passed slit
2) some momentum was imparted to the particle
3) I can measure the y-deflection and infer what momentum the
slit imparted to the particle.


-BUT the precise value of that imparted momentum is INDETERMINATE
until particle hits detector. Between the slit and detector, the particle
is entangled with the slit. The slit will not get the momentum-conserving
recoil until AFTER the particle hits the detector.

 

[OlderDan]

Then ALL ARPES results are crap! How do you explain that they explained the physics of material so well?

Zz.

I don't think ARPES is under attack here. I would like to read the article you cited early on

[1] T. Valla et al., PRL v.83, p.2085 (1999)

before engaging in a discussion of what it claims or does not claim. Is it available online? I have followed what links I can to other articles, but they are lacking the detail that I need to form my own opinion of what is being measured.

I'll go out on a limb here and suggest that the reason ARPES works is because it is not relying on one single particle measurement of momentum to reach any conclusion. It is measuring a sufficient number of particles to quantify the spread of momentum and energy values to say that within some limits of uncertainty that energy and momentum have a relationship that is characteristic of the material being investigated. Somewhere in the design of the experiment there are apertures that localize the particles involved, but those apertures are sufficiently wide so that there is still a beam of particles that can be characterized by some average momentum and energy.

All that I am troubled by is statements you have repeatedly made that suggest that HUP has nothing to do with limiting sumultaneous measurements of both position and momentum of a single particle, such as this one from post #34

Again, nothing from the experiment above prevents me from obtaining a definite value of position and momentum from a single measurement. The uncertainty in these values are not governed by the HUP, nor are they related.

In my opinion this statement is false, but that does not imply that I think ARPES is worthless.

 

[Antiphon]

I'll go out on a limb here and suggest that the reason ARPES works is because it is not relying on one single particle measurement of momentum to reach any conclusion.

This isn't right... Edit: (To clarify Dan, I mean I don't think this would bear on the
issue at hand. I'm not making a statement about how those experiments
are actually done)

Again, nothing from the experiment above prevents me from obtaining a definite value of position and momentum from a single measurement.

The two views in dispute are not at odds after the experiment.

After the data is collected and analyzed it is meaningful
to say as ZapperZ says the he has made accurate measurements of
position and momentum, and its true that the momentum was conserved
and that it was imparted to the particle by the slit.

But as I've mentioned above, in transit between the slit and screen, the
momentum imparted by the slit is not "known" even to the slit
until after the particle hits the detector. For this to be otherwise
implies either a violation of conservation of momentum, or a purely
classical trajectory. It is precisely at this time and place, between
slit and detector that the HUP is applicable.

Once we get to the point of analyzing the statistical data of many trials,
this uncertainty will manifest itself in the measurements as precisely the
statistical deviations which have been previously posted. But it would be
wrong to conclude from this that the uncertainty is only expressed
statistically and therefore applies only to an ensemble of many measurements.

Edit: Changed "screen" to "slit" for clarity.

 

[ZapperZ]

Sorry, I'm not familier with ARPES. But it's results are no doubt
valid because:

When the particle finally hits the detector and you detemine that it had
such-and-such y-compnent of the momentum, that's a correct summation
of the OUTCOME of the entire experiment.

That is, its not untrue to say
1) particle passed slit
2) some momentum was imparted to the particle
3) I can measure the y-deflection and infer what momentum the
slit imparted to the particle.


-BUT the precise value of that imparted momentum is INDETERMINATE
until particle hits detector. Between the slit and detector, the particle
is entangled with the slit. The slit will not get the momentum-conserving
recoil until AFTER the particle hits the detector.

Huh? The particle INTERACTED with the slit! Where is the momentum conserving problem here? Why do you think for any diffraction effects to be observable, the slit size must be comparable or smaller than the deBroglie wavelength of the particle?

Second, can you tell me WHAT experiment has actually measured, in ONE measurement, the superpostion of that particular observable corresponding to that measurement? If I measure position, can you show me what experiment that has EVER measured the superposition of position in ONE single measurement?

Third, I have a particle in a superposition of states of a|1> + b|2> + c|3>. I made a measurement and found the particle is in |3>. Are you saying my measurement is WRONG? Why? All I did was make a measurement on ONE particle and found it in one of the states it was in. What is wrong with that? I NEVER claim, nor does QM allow me to, measure ALL 3 states in one shot! If I decide to make another measurement of the identical system, I won't go nuts if I measure |2>, or |1>. In fact, I will detect all of them as I make more and more measurements.

Now do the single-slit experiment with ONE electron at a time passing through the slit and hitting a detector after the slit. I asked NOTHING about anything else other than (i) what's being measured at the slit and (ii) what's being measured at the detector. I say that the particle passed though the slit and thus, at the slit location. Then, when I detect it at the detector, it has a py momentum. Period. I go home and retire.

Zz.

 

[ZapperZ]

I don't think ARPES is under attack here. I would like to read the article you cited early on

[1] T. Valla et al., PRL v.83, p.2085 (1999)

before engaging in a discussion of what it claims or does not claim. Is it available online? I have followed what links I can to other articles, but they are lacking the detail that I need to form my own opinion of what is being measured.

I'll go out on a limb here and suggest that the reason ARPES works is because it is not relying on one single particle measurement of momentum to reach any conclusion. It is measuring a sufficient number of particles to quantify the spread of momentum and energy values to say that within some limits of uncertainty that energy and momentum have a relationship that is characteristic of the material being investigated. Somewhere in the design of the experiment there are apertures that localize the particles involved, but those apertures are sufficiently wide so that there is still a beam of particles that can be characterized by some average momentum and energy.

All that I am troubled by is statements you have repeatedly made that suggest that HUP has nothing to do with limiting sumultaneous measurements of both position and momentum of a single particle, such as this one from post #34

And I'll ask you this time what you mean by "simultaneous". If I show you an QM operation such as AB|u>, where A and B are operators, are you telling me that A and B are applied "SIMULTANEOUSLY" to |u>? Or does this mean A(B|u>), or does this make any difference? Or do you think that the particle that passed though the slit INSTANTANEOUSLY leave a mark on the detector? I did say the detector is at SOME distance way AFTER the slit, didn't I? Or is this not obvious? Or are you questioning that the mark on the detector has anything to do with the momentum operator?

I can no longer keep track what is what anymore.

As for ARPES, the electrons that in the material are already in superposition of energy and momentum. Yet, I can apply a classical trajectory to the DETECTED point on the CCD to assign and calibrate its momentum. In other words, the situation between the slit and the detector is identical to the situation between the metal AND the detector in ARPES. That is how the in-plane momentum is calibrated. The ONLY reason we do this over many electrons is to know the INTENSITY variation, not the "uncertainty" of each measurement. The uncertainty of a single electron hitting the CCD depends only on the resolution of the CCD detector AND the camera that's capturing the image, i.e. DETECTOR uncertainty.

I gave TWO e-print arxiv articles on Photoemission already even if you have no access to the other journal articles that I already cited.

Zz.

 

[Antiphon]

Huh? The particle INTERACTED with the slit! Where is the momentum conserving problem here?

It is precisely because if the particle has no precise momentum (which
it does not by the HUP), then the slit cannot experience a
momentum-conserving recoil unless that recoil is also indeterminate. This
is called "quantum entanglement".

Why do you think for any diffraction effects to be observable, the slit size must be comparable or smaller than the deBroglie wavelength of the particle?

I'm not saying that at all. Quantum entagnlement of the slit and particle
is how you conserve momentum in the whole system while not specifying
any numerical values for momentum before measuring it.

Second, can you tell me WHAT experiment has actually measured, in ONE measurement, the superpostion of that particular observable corresponding to that measurement? If I measure position, can you show me what experiment that has EVER measured the superposition of position in ONE single measurement?

It can't be done. That's part of my argument.

Third, I have a particle in a superposition of states of a|1> + b|2> + c|3>. I made a measurement and found the particle is in |3>. Are you saying my measurement is WRONG? Why? All I did was make a measurement on ONE particle and found it in one of the states it was in. What is wrong with that? I NEVER claim, nor does QM allow me to, measure ALL 3 states in one shot! If I decide to make another measurement of the identical system, I won't go nuts if I measure |2>, or |1>. In fact, I will detect all of them as I make more and more measurements.

Now do the single-slit experiment with ONE electron at a time passing through the slit and hitting a detector after the slit. I asked NOTHING about anything else other than (i) what's being measured at the slit and (ii) what's being measured at the detector. I say that the particle passed though the slit and thus, at the slit location. Then, when I detect it at the detector, it has a py momentum. Period. I go home and retire.

Don't retire, Zz.

Rather, consider the implications of what I wrote about quantum entaglement
in the post above.

Your position measurments created the momentum values that you have.
They are valid data. But those particluar numerical values were not
in existence as a feature of the particle until they had hit your detector.

The particle between slit and detector has no value of momentum.
Your measurement caused that value to manifest from amoung
a choice of many possible values. And the slit recoiled to conserve that
momentum only after your particle had hit your detector.

 

[ZapperZ]

It is precisely because if the particle has no precise momentum (which
it does not by the HUP), then the slit cannot experience a
momentum-conserving recoil unless that recoil is also indeterminate. This
is called "quantum entanglement".

HUH? A macroscopic object entangled with a quantum object? WHOA!

Your position measurments created the momentum values that you have.
They are valid data. But those particluar numerical values were not
in existence as a feature of the particle until they had hit your detector.

The particle between slit and detector has no value of momentum.
Your measurement caused that value to manifest from amoung
a choice of many possible values. And the slit recoiled to conserve that
momentum only after your particle hits your detector.

Maybe I'm stupid since I'm only a lowly experimentalist. But I only CARE about the fact that once the particle HITS the detector, I can calibrate it's transverse momentum py. Have I claimed to know MORE than this? Have I not, since the beginning, said that when I detect the signal on the CCD, I THEN, AFTER THE FACT, ONCE IT HAS OCCURED, MUCHO WAAAAY BEYOND DETECTOR HAS GIVEN SIGNAL, deduce its momentum?

And may I point out that ALL MEASUREMENTS are like this?

Zz.

 

[Igor_S]

Whoa! Hang on! You read it wrong! I said that the "position measurer" is the slit. The position on the DETECTOR/SCREEN corresponds to the "momentum measurer". I deduce the momentum of the particle from where it hits the screen. I do not use it to measure the ORIGINAL position, i.e. this is not the "x" in [x,p].

Look at the Photoemission references I gave you. When you see images in there that is similar to my avatar, note that these are RAW CCD images made by electrons hitting the "screen". The ONLY thing done to them is to calibrate the location of where they hit the screen with energy and momentum of each of those particles. The location on where the hit the detector is not the "position" operator.

I'm sorry if you had to repeat yourself, but I didn't get it right. OK, so CCD measures momentum of each particle (it is irrelevant to this discussion how it does it).

But, as antiphon stated:

I'm saying you claim a well-defined momentum exists between slit and
detector. It simply doesnt.

I too, agree that there is no numerical value (well-defined) for a momentum between the slit and CCD screen. When particle interacts with CCD, its wave function collapses into single-momentum state (before that, it has been in superposition) and that value is registered. Given enough measurements, CCD detector shows us composition of momenta what each of those particles had before hitting the screen. Before (hitting the screen) they were all "equal" in the sense that they had the same wave function. I think you (Zapper) agree on this, too.

It is precisely at this time and place, between
slit and detector that the HUP is applicable.

Ahhh... I can see now what you are referring to. In that space, wave function of that particle (or it's state) has changed (it is now more sharply peaked in position space, and more smeared in momentum space). The change in the wave function will be later detected (as more particles go throught), thus resulting in the confirmation of the HUP. You are claiming HUP "kicked in" at the moment wave function changed. I would say, this was kind of collapse of wave function. The "detector" (the slit) has definite width so it's not a collapse into single value, but it's no big difference (it's still "very rapid" change). Whatever you call this sudden change of wave function, it's (observable!) consequence is:

\left< \Delta x^2 \right> \left< \Delta p_x^2 \right> \geq \hbar^2 / 4.

This consequence is what I learned by the name of HUP. That's why I said you have to make many measurements to see HUP "in action".

 

[Antiphon]

HUH? A macroscopic object entangled with a quantum object? WHOA!

Yes, indeed.

Maybe I'm stupid since I'm only a lowly experimentalist. But I only CARE about the fact that once the particle HITS the detector, I can calibrate it's transverse momentum py. Have I claimed to know MORE than this? Have I not, since the beginning, said that when I detect the signal on the CCD, I THEN, AFTER THE FACT, ONCE IT HAS OCCURED, MUCHO WAAAAY BEYOND DETECTOR HAS GIVEN SIGNAL, deduce its momentum?

And may I point out that ALL MEASUREMENTS are like this?

You are certianly not stupid Zz. Everything you are doing is correct, and
you have been consistent and correct the whole time except for one very
tiny point.

Like I said, I figured out the exact source of the disagreement for this
thread.

You believe that the outcome of your expeirment accurately describes
the state of the system during the experiment. This is common sense,
but it's not true . Only when your experiement is finished
(i.e. particle is detected) does it make any sense to talk about the particular
momentum which the particle had.

The truth you must come to understantd is that while the particle was
in flight between the slit and detector, it did not posess the
momentum which you measured . It was your detector's
measurement that resulted in that particular momentum coming into
existence.

Edit: The entanglement comes into describe the slit's recoil. It can't
recoil a particular way if the particle has an indefinite momentum.

 

[ZapperZ]

You believe that the outcome of your expeirment accurately describes
the state of the system during the experiment. This is common sense,
but it's not true . Only when your experiement is finished
(i.e. particle is detected) does it make any sense to talk about the particular
momentum which the particle had.

The truth you must come to understantd is that while the particle was
in flight between the slit and detector, it did not posess the
momentum which you measured . It was your detector's
measurement that resulted in that particular momentum coming into
existence.

But this is OBVIOUS!

The question is, did what I just measured as any resemblence to anything MEANINGFUL. Did the momentum I measured corresponds to ANY momentum anywhere. Did the momentum and energy that I measured in ARPES can resemble the momentum and energy of the electron while it was in the solid? I claim that they do! Why? Because what I measured corresponds to the THEORETICAL description of the band structure of the electrons IN THE SOLID. I am not measuring something DISCONNECTED with anything before a measurement, even when it is in a superposition. I may not measure the superposition itself in a single shot, but I can make REPEATED measurement to detect such superposition! Thus, my single measurement reflects a component of the system, and reflects this ACCURATELY AND CORRECTLY.

And if you care to look at (i) either Kittel or Ashcroft&Mermin solid state text and look at the conduction band and then (ii) compare that with the photoemission result on a "typical" metal surface (summary on Mo(110) surface state in http://arxiv.org/abs/cond-mat/0507653), I will tell you that many solid-state physics students GASP that they can actually SEE the E vs k dispersion curve from the RAW data of an experiment!

Zz.

 

[Antiphon]

But this is OBVIOUS!

I'm tremendously relieved!

The question is, did what I just measured as any resemblence to anything MEANINGFUL. Did the momentum I measured corresponds to ANY momentum anywhere.

Indeed it does.

Did the momentum and energy that I measured in ARPES can resemble the momentum and energy of the electron while it was in the solid? I claim that they do! Why? Because what I measured corresponds to the THEORETICAL description of the band structure of the electrons IN THE SOLID.

No being an expert on your technique, I'm sure you are correct.

I am not measuring something DISCONNECTED with anything before a measurement, even when it is in a superposition.

Disconnected is the wrong idea. Correlated but not caused is better.

I may not measure the superposition itself in a single shot, but I can make REPEATED measurement to detect such superposition! Thus, my single measurement reflects a component of the system, and reflects this ACCURATELY AND CORRECTLY.

I agree completely.

And if you care to look at (i) either Kittel or Ashcroft&Mermin solid state text and look at the conduction band and then (ii) compare that with the photoemission result on a "typical" metal surface (summary on Mo(110) surface state in http://arxiv.org/abs/cond-mat/0507653), I will tell you that many solid-state physics students GASP that they can actually SEE the E vs k dispersion curve from the RAW data of an experiment!

I am convinced you are quite right about that. The quantum measurements
you make do accurately reflect the materials under study.

:!)

Edit: Igor, yes. I agree with you too, especially about the last part of your post.

 

[OlderDan]

I want to break this into two parts, because I don't want there to be any confusion about what I am questioning

It is precisely at this time and place, between slit and detector that the HUP is applicable.

Once we get to the point of analyzing the statistical data of many trials, this uncertainty will manifest itself in the measurements as precisely the statistical deviations which have been previously posted. But it would be wrong to conclude from this that the uncertainty is only expressed statistically and therefore applies only to an ensemble of many measurements.

This I agree with, and it is the central point I have been trying to make. The HUP applies to every single particle that traverses the gap between the slit and the screen. The particle does not have a definite momentum in this region, and it cannot because it has been localized by the slit. HUP demands that the particle not be in a momentum eigenstate from the moment this localization takes place and at least until some future interaction.

The two views in dispute are not at odds after the experiment.

After the data is collected and analyzed it is meaningful to say as ZapperZ says the he has made accurate measurements of position and momentum, and its true that the momentum was conserved and that it was imparted to the particle by the slit.

But as I've mentioned above, in transit between the slit and screen, the momentum imparted by the slit is not "known" even to the slit until after the particle hits the detector. For this to be otherwise implies either a violation of conservation of momentum, or a purely classical trajectory.

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? 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.

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.

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. 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?

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.

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 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.

If you allow an ensemble of particles to pass through the slit you create a "beam". Classically, if the slit were imparting varying amounts of transverse momentum to the particles, each particle (assumed non-interacting) in the beam would be confined to one linear path within the beam. We could say that an arrival at a point on the screen limited the direction of arrival to a narrow wedge extending back from the point of arrival to the edges of the slit. In QM we cannot do that. The beam in QM is a spreading wave packet within which each particle is non-localized until it is position detected. When it is position detected by the screen, we cannot infer anything about where it might have been detected if the screen had been placed somewhere else, except to say it had to be somewhere in the beam. I will elaborate on this.

Suppose we did not know about the slit and we detected a hit on the screen. What would we know about the momentum of what hit the screen? Nothing more than perhaps which side of the screen it came from. If our screen had detectors that responded differently to particles of different energies, then we could say we know where the particle hit, and how much energy it had, so we could deduce the magnitude of the momentum, but we still would not have a measurement of the direction of arrival. Now the slit comes into the picture. If we can say that the particle came through the slit and followed a straight line trajectory from the slit to the screen, then we can say we know the direction of the momentum. Classically we can surely do that, but can we do it in QM? I think not. 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.

To be more specific, let y be the transverse coordinate in a single slit experiment. One particle comes through a narrow slit centered at y = 0. At a distance L from the slit in the forward direction, say I detect the particle at y = a. Classically I can infer that if I had placed my screen at a distance L' from the slit then I would have detected that same particle at position y = a(L'/L). In QM, if I had placed the screen at some other location it might have detected the particle at any y position permitted by its wave function. If you imagine that the particle actually had an "undetermined" position at distance L/2 of y = -a/2, it might still arrive at distance L with a position y = a. In fact, the particle could arrive at y = a at distance L from any prior location consistent with the evolution of its wave function. This leads me to conclude that the only way to measure the momentum of the particle when it hits the screen is to measure the direction of arrival. 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 momentum states and I detect its position without measuring its direction of arrival, then I can only conclude that it arrived with the distribution of momentum states included in the superposition. I have not selected anyone momentum state.

So how can I use a position measurement to measure momentum? I can do it by placing my detector in a position such that the only particles that it can possible detect are particles that have a narrow spectrum of momentum. For example, if instead of a single slit we had multiple slits we would create a very different situation. Although the individual slits are extremely narrow, we are not forcing a particle through anyone of them. The more slits the better because the more of them you have, the better you can associate the possible subsequent positions of the particle with its momentum distribution. What you have accomplished is to create beams originating from the same aperture (not too narrow) all going in different directions. A beam is characterized by a width that is going to spread depending on its momentum spectrum. If you start with a large enough aperture, the momentum spectrum of a beam going in some direction can be very narrow.

This is what happens when you use a diffraction grating spectrometer to observe the emission spectra of an element like hydrogen. If you allow a beam of light from hydrogen gas discharge to pass through a huge number of extremely narrow slits, out comes a bunch of collimated beams traveling in different directions. The incoming beam is not monochromatic; it is really a spatial mixture of several different beams each characterized by a well defined momentum. At the grating, each beam has picked up a substantial transverse momentum, but with a corresponding change in the forward momentum to conserve energy (wavelength). More importantly, from a QM perspective each beam is a very slowly spreading wave packet with a very narrow momentum distribution permitted by the fact that the initial aperture, while small enough to produce a collimated beam, is not small enough to cause significant aperture spreading. If you make detections that are far enough away from the grating, you can infer the momentum quite accurately because the only way the particle could get to that detection location is by having a momentum in the very narrow spectrum of that beam. You have spatially separated the wave packets of particles of different energy and average momenta. You cannot say that detections on the left side of one beam have different momentum than detections on the right side of that beam. What you can say is that any detection associated with that beam has the momentum distribution characteristic of that collimated beam.

If instead of a gas discharge tube you use a white light source, you create a continuous spread of beams. No matter how far away you look back toward the grating, you will never completely separate the beams, but you do get better separation as you move farther away. With a small sensitive detector you can be quite selective about the range of momenta you permit to hit the detector, and you can improve the resolution by moving farther away.

Diffraction gratings are of course not the only things that can spatially separate overlapping beams. A beam of charged particles will interact with an electromagnetic field. Classically, a uniform magnetic field puts each charged particle into a circular trajectory whose radius depends on its momentum (its velocity), so half a circle away particles of the same mass with different energies are separated, or particles with the same velocity and different masses are separated. For each of them we can draw a circular trajectory back to the point of entry into the field. In QM, each of the momentum components of a beam will be affected by the field. If the incoming beam is a mix of particles with different momenta, and if we do not force them through a narrow aperture at entry that introduces significant spreading of their wave packets, then position measurements half a circle away can be used infer their momentum at entry into the field. Mass spectrometry of course relies on these final position measurements to infer momentum at field entry. But if at the point of entry you forced one particles through a narrow slit you would cause spreading of its wave packet in the field region and you would no longer be able to say that it followed a circular trajectory from the point of entry to the point where it hit the detector. Its evolution from the point of entry to the detector is known only as a position probability density. If I had detected the position of that same particle after an assumed 90 degree turn then I might have measured a very different circular path radius than the one I measured after an assumed 180 degree turn. If I forced it through a narrow slit, I cannot assume I know the direction it was heading when I detected it. Even classically you would be in trouble if you did not know with considerable certainty the direction of the particle at field entry because then you would not know where to locate the center of curvature of the classical trajectory. Hence the usual velocity selector at the front end of the mass spectrometer. The last thing I would want to do is introduce dispersion by using a too-narrow entrance slit.

I managed to find information about ARPES experiments and apparatus that I did not want to speculate too much about. 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.

http://www.physics.ubc.ca/~quantmat/ARPES/PUBLICATIONS/Reviews/ARPES_intro.pdf

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.

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. 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.

 

[jackle]

...I am also still confident that in the single slit scenario a hit on the screen does not constitute a momentum measurement...

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 have noticed that there are loads of different brain heads that post here, who each look at things very differently but who all seem to know their stuff. It is hard for me to work out the difference between interpretation and fact sometimes.

 

[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.