Heisenberg imcertainty principle (get it!)

[mrfeathers]

I dont understand why it is so hard to find the exact position and velocity of orbiting electron. And also, why would we want to know it, if it is always moving? im not trying to disprove it or anything, so dont make fun of me, i am an uneducated peon

[buddingscientist]

whats your idea of finding both the exact position and velocity of the electron?

how do you find the location of your keyboard? shine light on it (lamp or diffuse sunlight through the window) what happens if you try do the same for something tiny like an electron ?

[mrfeathers]

you should do that then, just prove the heisenberg uncertainty therom but just looking at the orbiting electrons with your own eyes

[buddingscientist]

when you look at your keyboard, the photons of light that 'interact' with it (and ten hit our eyes and give us what we percieve as colour) are much much smaller than the keyboard.
but if we are trying to see an electron using a photon..
the wavelength ("size") of an photon is alot larger than that of an electron, what implications does this have?

[Dr.Brain]

For complete understanding of Uncertainity Principle , check out :

http://www.doxlab.co.nr/

TOPIC#2 on the above site is Heisenberg's..

[ZapperZ]

For complete understanding of Uncertainity Principle , check out :

http://www.doxlab.co.nr/

TOPIC#2 on the above site is Heisenberg's..

I would not recommend the site listed above. It perpetuates the misconception of the HUP that I have written about[1], that it is the uncertainty in a single measurement. I have seen this mistake repeated several times within the past week on here.

The HUP is NOT about the uncertainty in INSTRUMENTATION or measurement. One can easily verify this by looking at HOW we measure certain quantities. It is silly for Heisenberg to know about technological advances in the future and how much more accurate we can measure things. This is NOT what the HUP is describing. The HUP is NOT describing how well we know about the quantities in a single measurement. I can make as precise of a measurement of the position and momentum of an electron as arbitrary as I want simultaneously, limited to the technology I have on hand. I can make improvements in my accuracy of one without affecting the accuracy of measurement of the other.

What the HUP is telling you is the difference between a classical system and a quantum system. In a classical system, if you have a set of identical initial condition, and you measure ONE observable, and then you measure another observable, you will continue to get the SAME value of that 2nd observable everytime you measure the same value of that 1st observable. The more accurate you measure the 1st observable, the more accuract you can predict the value of the 2nd observable the next time you want to do such a measurement. The only limitation to how accurate you can determine these observable is the limitation to your measuring instruments. But these limitations do NOT scale like the HUP. You don't make one worse as the other one becomes better, because these are technical issues and are not related to one another.

On the other hand, in a quantum system, under the IDENTICAL initial conditions, even if you measure a series of identical values for the 1st observable, the 2nd observable may NOT yield the identical result each time. In fact, as you narrow down the uncertainty of the 1st observable, the 2nd observable may start showing wildly different values as you do this REPEATEDLY. Therefore, unlike the classical system, your ability to know and predict what is going to be the outcome of the 2nd observable goes progressively WORSE as you improve your knowledge about the 1st observable!

Again, it has NOTHING to do with the uncertainty in a SINGLE measurement! It doesn't mean that if you measure with utmost accuracy the position of an electron, that that electron momentum is "spread out" all over the place. This is wrong! I can STILL make an accurate determination of that electron's momentum - only my instrument will limit my accuracy of determining that. However, my ability to know what its momentum is going to be the NEXT time I measure it under the idential situation is what is dictated by the HUP!

Zz

[1] [11-15-2004 09:26 AM] - Misconception of the Heisenberg Uncertainty Principle

[matness]

time is also an observable so why do we always take Δt= 0 while
Δx*Δp >= hbar/2 ?

[ZapperZ]

time is also an observable so why do we always take Δt= 0 while
Δx*Δp >= hbar/2 ?

Who is this "we"?

You will note that in typical school problems, you often work with a solution to the time-INDEPENDENT solution. It doesn't mean that the uncertainty in the time period is zero. It means that for that case, it is irrelevant since the description does not contain any time dynamics.

Zz.

[Nomy-the wanderer]

I've a simple answer, hope u won't consider it naive...

Most of the working theories right now weren't based on solid proofs but because we needed them, and some observations needed explanations...Theorists make theories, without confitmations, but we assume they r correct untill they r proven wrong, technology gives us the chance to make sure that we r on the right track, we can't yet find the electron and the uncertainty principle is what really works right now...

See u when they find out something strong that would give us the opportunity to observe electrons..

[ZapperZ]

I've a simple answer, hope u won't consider it naive...

Most of the working theories right now weren't based on solid proofs but because we needed them, and some observations needed explanations...Theorists make theories, without confitmations, but we assume they r correct untill they r proven wrong, technology gives us the chance to make sure that we r on the right track, we can't yet find the electron and the uncertainty principle is what really works right now...

See u when they find out something strong that would give us the opportunity to observe electrons..

What are "solid proofs"? Would the statment that says "IT WORKS" be considered as "solid proof"? How about if I point to you your modern electronics? Would that be considered as "solid proof"?

Most people forget that the HUP is a CONSEQUENCE, not the origin, of quantum mechanics! To find a problem in the HUP is to find a problem in QM. And unless people also forgot about the centenial year of QM in 1999, let me remind you that there was an almost universal acclaimed by physicists that QM is THE most successful theory SO FAR in the history of human civilization!

This means that if you think QM does not have "solid proofs", then other parts of physics suffer from even a worse level of lacking of solid proofs!

Please keep in mind that NO part of physics is considered to be accepted and valid until there are sufficient experimental/empirical agreement! In fact, many theories and ideas originally came out of unexpected experimental observation in the first place!

There is no lacking of "solid proofs" for QM, and the HUP. Why this is even brought up here, I have no idea.

Zz.

[matness]

"i" was thinking these as you said , until i see the word 'simultaneous' for measurements in the defn of HUP. if we say nearly simultaneous then i think there will be no problem(i hope so...)

Also there are different explanations for HUP, and maybe this the problem about understanding it. at first i was thinking i get it , but it didnt take a long time for me to confuse (because i am a beginner only)
Zz 's article is very helpful but if anyone can send a sketch of proof for HUP it will be more clear

thanks
n

[ZapperZ]

"i" was thinking these as you said , until i see the word 'simultaneous' for measurements in the defn of HUP. if we say nearly simultaneous then i think there will be no problem(i hope so...)

Also there are different explanations for HUP, and maybe this the problem about understanding it. at first i was thinking i get it , but it didnt take a long time for me to confuse (because i am a beginner only)
Zz 's article is very helpful but if anyone can send a sketch of proof for HUP it will be more clear

I don't understand. You want a "sketch of proof" for the HUP? What is this?

I think every student has either done, or seen the diffraction from a single slit. To me, this is a VERY clear example of the HUP! It just happens that we typically use wave description of light to account for such effects. But with the photon picture, the identical diffraction pattern can be directly obtained and the spreading is a direct consequence of the HUP!

More? The deBoer effect that is very pronounced in noble gasses is a direct consequence of the HUP. This leads to a correction to the internal energy (and thus, the specific heat capacity) of the gasses at very low temperatures. Only via taking into account such corrections can one obtain the experimental values!

But I think people pay waaaaay too much attention at disecting the consequences and forgetting the principles that CAUSE such consequences. The fact that this came out of "First Quantization" principle of QM that is based on [A,B] operations of two non-commuting observables is less understood by many who do not understand the formalism of QM. This is a crucial part of elementary QM with which a whole slew of consequences are built upon!

Zz.

[matness]

what i wonder is the mathematical part :

[A,B]= C --> ΔA * ΔB >= |<C>|/2

is it just a thm about standart deviation?

[seratend]

I dont understand why it is so hard to find the exact position and velocity of orbiting electron. And also, why would we want to know it, if it is always moving? im not trying to disprove it or anything, so dont make fun of me, i am an uneducated peon

The second part answers to the first part of your post.

In QM, we may define the position of a particle. We may also define the momentum of a particle. However momentum is not the velocity of the particle. Many people in this forum always mix the momentum with the velocity (due to their equality for average values: i.e. Erhenfest Theorem) and tend to make incorrect deductions.
QM tells one thing: we cannot associate a classical path to a particle, hence we cannot define a couple (position, velocity) to the particle.
The HUP property applied to the (position, momentum) observables just highligh this fact: we cannot find a particle where both position and momentum have "defined" values equal to their mean values (i.e. through the Erhenfest Theorem, they have a classical path if is the case).

Seratend.

[seratend]

what i wonder is the mathematical part :

[A,B]= C --> ΔA * ΔB >= |<C>|/2

is it just a thm about standart deviation?

Yes.

Seratend.

[jackle]

I can make as precise of a measurement of the position and momentum of an electron as arbitrary as I want simultaneously, limited to the technology I have on hand.

What sort of technology do we have to achieve this?

[quetzalcoatl9]

The second part answers to the first part of your post.

In QM, we may define the position of a particle. We may also define the momentum of a particle. However momentum is not the velocity of the particle. Many people in this forum always mix the momentum with the velocity (due to their equality for average values: i.e. Erhenfest Theorem) and tend to make incorrect deductions.
QM tells one thing: we cannot associate a classical path to a particle, hence we cannot define a couple (position, velocity) to the particle.
The HUP property applied to the (position, momentum) observables just highligh this fact: we cannot find a particle where both position and momentum have "defined" values equal to their mean values (i.e. through the Erhenfest Theorem, they have a classical path if is the case).

Seratend.

my (rather naive) understanding of it is that in a classical system:

xp - px = 0

but accord to HUP:

xp - px \neq 0

so that measuring the position and then measuring the momentum, is not the same as measuring the momentum and then measuring the position. infact, they will always differ by \frac{ih}{2\pi}

position and momentum are not seen as real values, but as non-commutative operators. you can derive the schrodinger wave equation in a straightforward manner from this.

(this is one interpretation).

[Kruger]

so that measuring the position and then measuring the momentum, is not the same as measuring the momentum and then measuring the position. infact, they will always differ by

No, that's not correct. The HUP doesn't says something about only one simultaneoulsy measurment. It says something about a serie of simultaneously measurements (always the same conditions). You see?
If you make 1000 simultaneoulsy measurments (position and momentum) and you measure the position at each experiment exactly then you will get at each measurement of momentum a completely different value.

[Igor_S]

Many people in this forum always mix the momentum with the velocity (due to their equality for average values: i.e. Erhenfest Theorem) and tend to make incorrect deductions.

Umm, how can momentum be equal to velocity ? Ehrenfest theorem is about equality of average QM momentum, which is defined as -i \hbar \vec \nabla and classical momentum m \vec v. Or generally it shows that average values of QM operators are equal to corresponding quantities in classical mechanics (I guess that's what you had in mind).

---
quetzalcoatl9,

HUP actually does say \left< (\Delta p_x)^2 \right> \left< (\Delta x)^2 \right> \neq 0. What you wrote, are commutation relations for operators, not a HUP (however, it's used in derivation of HUP). And you cannot get deltas by taking ONE particle. Let's say you have an instrument which determines position and momenutum up to 10 decimal places [in some units]. And let's say measurement of momentum of electron always gives one value, eg. 1.0000000001 [in some units]. Then you measure it's position 1st time and get let's say 0.0000000044 [some units]. A set of repeated measurement of position (with momentum set to 1.0000000001) will get you just random results, like 50.3243243212, 0, 13.1313131313, -400000.0000000001, etc. (but each of these measurements will have high precision, and that's the one that depends on the instrument itself!). However, calulating the average of the square of deltas of random numbers like that (you see, I HAVE to have more than one measurement to do average!), gives us a big number. This number appears in HUP.

When I let particle momentums have some distribution of let's say between 1.0 and 1.1 , the numbers I get for position will not be so random - they will have a peak value around some number. If I don't control momentum at all, but let particles pass a very small hole (that way I'm controling position) and measure momentum afterwards, I will get random results.

Hope this helps! :smile:

[seratend]

Umm, how can momentum be equal to velocity ? Ehrenfest theorem is about equality of average QM momentum, which is defined as -i \hbar \vec \nabla and classical momentum m \vec v.

Ok, let's explain the "due to their equality for average values" in my previous post.
<P>=m.d<X>/dt= m<V> if no em field (case H=p^2/2m+V(q)).
=> implictly assuming m=1 units, we have <P>=<V> QED.

(I thought it was clear enough, but your post showed I was wrong with this assumption, now I hope it is clearer).

However, for a given relation V=dX/dt on operators (e.g. V=P/m), we have the eigenvalue relation v=dx/dt iff [V,X]=0 (if the operators are sufficently "gentle").

If V and X have not the same eigenbasis (in other words they do not commute: [V,X]=/=0) => the relation v=dx/dt is no more valid for the eigenvalues => we cannot associate a classical path to a particle.

HUP just reflects this fundamental property of operators.

Seratend.

[Igor_S]

Yes, it crossed my mind later that you assumed m=1, but in physics it's unusual to equal momentum and velocity (or it's done very rare, because mass is not fundamental constant).

I understand that in math, it's just a number, so who cares ? :biggrin:

[jackle]

I can make as precise of a measurement of the position and momentum of an electron as arbitrary as I want simultaneously, limited to the technology I have on hand.


What sort of technology do we have to achieve this?

I'd still really like to know the answer to this because it contradicts what I have been told by a trusted source. It is obviously a very fundamental principle of reality in our universe, so I think it is vital to know the facts. Ideally, I'd like a name of an established experiment where physicists can measure a complimentry pair simultaneously to an accuracy that demonstrates what you are saying.

Thanks.

[jackle]

Oh, by the way, you are also a trusted source, which is why I am being so persistent.

[ZapperZ]

I'd still really like to know the answer to this because it contradicts what I have been told by a trusted source. It is obviously a very fundamental principle of reality in our universe, so I think it is vital to know the facts. Ideally, I'd like a name of an established experiment where physicists can measure a complimentry pair simultaneously to an accuracy that demonstrates what you are saying.

Thanks.

But think about this (based on what I described in the single-slit experiment).

1. How well I know the location of a photon depends on how wide a slit I make, no? The smaller the slit, the more I know about where that photon was when it passed by it.

2. When it passed by the slit, what is its transverse momentum perpendicular to that slit? Unless its momentum changed between the slit and the detector, then I can make the assumption that this momentum remained the same between the moment it passed through the slit and the moment it hits the detector. All I need to do is figure out WHERE on the detector it hits. Then, using simple geometry, I know it's momentum in that direction. How well I determine that momentum depends on the resolution of my detector, i.e. how well can I determine where it hits the detector. The larger the number of pixels on my CCD camera, for example, the finer I can determine this location.

Both 1 and 2 allow me to determine the position and momentum of an INDIVIDUAL photon (or electron, or neutron) to arbitrary precision (i) independent of each other and (ii) depend entirely on the technology of the measurement apparatus. The HUP doesn't kick in here! The HUP kicks in on the subsequent photon IF I try to make a prediction on its momentum under the SAME slit size! The HUP also kicks in if I repeat this measurement many times till I get a statistical spread on the momentum value with a fixed slit width.

There are many state-of-the-art apparatus that use techniques. Photoemission spectroscopists are familiar with one - their hemisphrical detector, such as the Scienta SES electron analyzer, allows for the E vs k measurement simultaneously in one shot with electrons passing through a slit (see my avatar).

Zz.

[jackle]

I'll do some reading.

Thanks

[Antiphon]

ZapperZ-

I agree with what you say about the HUP having nothing to do with the accuracy of measuring devices.

However I strongly disagree with your interpretation of HUP. It very much
is about the impossibility of determining simultaneous position and momentum
in a single measurement.

You can easily determine both position and momentum with arbitrary
accuracy as long as you don't try to do it in a single measurement.
Here's a prescription for doing this.

1) Measure a particle's position with arbitrarily small uncertainty
2) Measure the same particle's position again with arbitrarily small uncertainty

The momentum the particle had between the two places is the
distance divided by the time.

You now know the position of the particle at two points with very
little uncertainty and you also know the momentum which the particle
had between (and arbitrarily close to) the two measurements. What
you cannot know is the momentum of the particle just after an accurate
position measurement.

It's the very act of the second measurement which renders the
momentum of the particle unkown just after that second measurement.

[ZapperZ]

You now know the position of the particle at two points with verylittle uncertainty and you also know the momentum which the particle
had between (and arbitrarily close to) the two measurements. What
you cannot know is the momentum of the particle just after an accurate
position measurement.

It's the very act of the second measurement which renders the
momentum of the particle unkown just after that second measurement.

Come again?

Unless I misread your description, you are implying that from the moment the particle passes through the slit and BEFORE it gets to the detector, the momentum of the particle is unknown or could CHANGE and is not what I measure at the detector. Is this true?

If this is true, then the momentum that I'm measuring CAN be dependent on where I put the detector from the slit. If I put it at 1 meter after the slit, that should give me a different momentum value than when I put it 12 meters after the slit. Last time I checked, this is not the case. Based on this, I can deduce that the momentum of the particle after it left the slit is the same no matter where I put the detector. Thus, I'm measuring the lateral momentum of that particle the instant it left the slit when its position confinment is applied.

Applying the HUP for ONE single measurement is absurd. I will remind you of the DEFINITION of the uncertainty of an observable in QM, which is:

(\Delta A)^2 = <A^2> - <A>^2

Now what are the averages of A and A^2 when you have just ONE measurement? And what is the uncertainty in THAT measurement?

Zz.

[Antiphon]

Unless I misread your description, you are implying that from the moment the particle passes through the slit and BEFORE it gets to the detector, the momentum of the particle is unknown or could CHANGE and is not what I measure at the detector. Is this true?


Here's what I'm saying. An accurate position determination by definition
peaks the wavefunction \Psi(x,t) at point in space.

Mathematically this renders the momentum indefinite. There IS no one
unique value of momentum attributable to the particle after this
accurate position measurement.

In order to have a space-compressed wave function it will have a
momentum-space representation which is a superposition of a broad
spectrum of momenta. The actual (definite) momentum of the particle
will only come into existence at a future time IF a momentum measurement
is made.

This is the true meaning of the HUP.

[jackle]

I didn't get anywhere with my reading but I was able to find web references to Photoemission spectroscopists, hemisphrical detectors and Scienta SES electron analyzers.

I was a bit worried that if you took a calculation approach in general for obtaining momentum, you might get results that are never measured in practice. For example, if an electron tunnels through a barrier, it might seem to have travelled faster than light to get to the other side under some circumstances? Dunno.

[ZapperZ]

Here's what I'm saying. An accurate position determination by definition
peaks the wavefunction \Psi(x,t) at point in space.

Mathematically this renders the momentum indefinite. There IS no one
unique value of momentum attributable to the particle after this
accurate position measurement.

In order to have a space-compressed wave function it will have a
momentum-space representation which is a superposition of a broad
spectrum of momenta. The actual (definite) momentum of the particle
will only come into existence at a future time IF a momentum measurement
is made.

This is the true meaning of the HUP.

But you have not addressed what I have pointed out. And what exactly do you mean by "definite"? Do you mean that in ONE shot, or do you mean my ABILITY to predict what the range of values would be IF I were to actually MAKE th measurement?

Take a free particle. Write down the wavefunction for that particle as a single plane wave. It can have a very DEFINITE momentum, or k (let's put a delta function there). However, if you try to PREDICT where it is located, you have a HUGE uncertainty. However, does this mean that I CANNOT measure the position at a given instant? There's nothing to prevent me from measuring a position of the particle as accurately as I want. Let's say I found it to be at x1 at time t1. If I prepare the IDENTICAL situation again, at the identical time t1, do you think I'll get x1 again? Classical mechanics says YES. Quantum mechanics says NO. This is because the situation has a very large uncertainty. The particle could be at x2 that is hugely different than x1. Your ability to know where the particle is is GONE because the momentum is so well-defined! THIS, is what is meant by the HUP. It isn't my ability to measure a single measurement.

Again, use the DEFINITION of the HUP. It is a series of measurements and one's ability to predict where the next one is going to be. It is not about ONE single measurement.

Zz.

[ZapperZ]

I didn't get anywhere with my reading but I was able to find web references to Photoemission spectroscopists, hemisphrical detectors and Scienta SES electron analyzers.

I was a bit worried that if you took a calculation approach in general for obtaining momentum, you might get results that are never measured in practice. For example, if an electron tunnels through a barrier, it might seem to have travelled faster than light to get to the other side under some circumstances? Dunno.

Er... you are forgetting that we use tunneling spectroscopy and photoemission spectroscopy to MEASURE the properties of various materials. If what we interpret out of those measurements are "never meausred in practice", then these materials that we are studying, when PUT into work in your electronics, would NEVER perform the way we have characterized them! In case you did not know, the EARLIEST confirmation of the correct band structure of semiconductors came from photoemission measurements! I don't think I need to explain further the importance of semiconductors AND the knowledge of their band structure, do I?

Zz.

[Antiphon]

But you have not addressed what I have pointed out.


I would if I could. But I'm only referring to one free particle and one measurement.
I'm not sure what your slits and such are all about.


And what exactly do you mean by "definite"? Do you mean that in ONE shot, or do you mean my ABILITY to predict what the range of values would be IF I were to actually MAKE th measurement?


By definite I mean that that outcome of the observation has a specific
and predictible numerical value.

I mean in one shot, if you localize a particle's position then you have obliterated
any hope of predicting a narrow range on the expecation value for the momentum operator.

You can always predict the range of values- and in this example the range on the
momentum is unlimited. It could be anything WHEN you finally get around to making
the measurment.


However, does this mean that I CANNOT measure the position at a given instant? There's nothing to prevent me from measuring a position of the particle as accurately as I want.


You almost have it Zz. You CAN measure it- you just can't PREDICT it. And as soon
as you make your very accurate position measurement you no longer have a definite
mometum- no more plane wave and no more definite momentum.

-All in a single measurement. THAT's the HUP.

[jackle]

...when PUT into work in your electronics, would NEVER perform the way we have characterized them...

I've often wondered why my computer keeps crashing in a very unpredictable way. I always put it down to software...

[ZapperZ]

I would if I could. But I'm only referring to one free particle and one measurement.
I'm not sure what your slits and such are all about.



By definite I mean that that outcome of the observation has a specific
and predictible numerical value.

I mean in one shot, if you localize a particle's position then you have obliterated
any hope of predicting a narrow range on the expecation value for the momentum operator.

You can always predict the range of values- and in this example the range on the
momentum is unlimited. It could be anything WHEN you finally get around to making
the measurment.



You almost have it Zz. You CAN measure it- you just can't PREDICT it. And as soon
as you make your very accurate position measurement you no longer have a definite
mometum- no more plane wave and no more definite momentum.

-All in a single measurement. THAT's the HUP.

If you have read my very early treatment of the single slit measurement, you would have noticed that I emphasized the ABILITY TO PREDICT several times!

Again, I have said repeatedly, that there is NOTHING to prevent you from making as accurate of a measurement of a SINGLE value of position and momentum. Period. I have said, again repeatedly, that one's ability is only limited by technology - how small a slit one can make, and how many fine pixels on the detector that the photon or electon hits! These are instrumental uncertainty, NOT the uncertainty in the HUP! [I feel as if I'm repeating this forever!]

But after one has made ONE measurement set (particle passing through slit, hits detector, so one has position and momentum), THEN, the very next one, if the identical sitation occurs and the particle passes through the slit, one's ability to PREDICT where the next one is going to hit the detector depends VERY MUCH on how small the slit is (i.e how small Delta(x) is!). If one does this a gazillion times, one will know Delta(x) by the size of the slit, but Delta(p) will be VERY large from the statistics alone if Delta(x) is small.

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. If you are still claiming that they are, then tell me how my ability to change the width of the slit affects the density of the number of pixel on my CCD plate at the detector.

Zz.

[ZapperZ]

I've often wondered why my computer keeps crashing in a very unpredictable way. I always put it down to software...

Then if you think we have characterized it wrong, you should not fly commercially, drive your car, seek medical treatment, believe in the value of h and e, etc.

Zz.

[jackle]

Then if you think we have characterized it wrong, you should not fly commercially, drive your car, seek medical treatment, believe in the value of h and e, etc.

Zz.

I was just kidding. Your doing a great job!
:smile:

[Antiphon]

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. If you are still claiming that they are, then tell me how my ability to change the width of the slit affects the density of the number of pixel on my CCD plate at the detector.


For the most part I agree with everything you say Zz. Let me be extremely
clear with the simplest example.

As you say, you certainly can measure position and momentum at the
same time and get definite values. And the uncertainties in the accuracy
would be related to your measuring technology exactly as you say.

Suppose you have a single free particle in a definite momentum state
(planewave, no localized position of the wavefunction.) Then you can
measure the momentum all day long and get the same definite value
each time you measure it without ever disturbing that momentum. It
is a predictable and definite value.

But if sometime along the way you also attempt to localize the particle
(that is, make a position measurement) then that position measurement
will introduce an uncertianty into any subsequent momentum measurement.

The uncertainty is proven out by the second measurement. But the
outcome (of a decreased predictability in the mometum value) was sealed
the instant you made the first position measurement.

In fact, the HUP applies even to wave packets that are not being
measured at all, like a minimum-uncertainty gaussian becasue it is a
purely mathematical consequence of the fact that specific positions are
built up of planewaves of all possible momenta, and a specific momentum
is a planewave which covers all of space (no particular position).

[ZapperZ]

But if sometime along the way you also attempt to localize the particle
(that is, make a position measurement) then that position measurement
will introduce an uncertianty into any subsequent momentum measurement.

In what form is the uncertainty introduced in a SINGLE measurement of the momentum? I have a very thin slit. At some time, your free particle passed through it. I say "Ah ha! At time t1, there was a particle going through position x1, and my uncertainty in the position is +- width/2!"

But being smart, I also put a CCD screen behind the slit. I can record, to ARBITRARY ACCURACY limited by my CCD technology, where the particle hit the detector. The lateral position (in the direction perpendicular to the slit) of where the particle hit tell me the momentum in this direction. The uncertainty in this momentum depends only on the uncertainty in determining where the electron hits the CCD. In fact, the higher the resolution of the CCD, the higher the accuracy with which I can determine the position the particle hits the detector. I have made measurement in which the accuracy is has high as 2 pixels on a CCD! This has zero to do with the width of the slit or the HUP. And voila, I have made a DEFINITE single measurement of position x AND momentum p_x of your particle.

I have just done what you said cannot be done.

In NONE of these have I said anything about "predictability". All I care about is the question "can I make as an accurate of a position and momentum measurement of a single particle?"

The answer is :yes. This is because that question depends on the technology of detection.

But the question: "if I make the position measurement VERY, VERY accurate (very small slit), then can I predict with equal accuracy where there the particle will hit the CCD and thus, determine it's transverse momentum?" The answer is NO.

These two are DIFFERENT QUESTIONS under different circumstances. You cannot say I cannot make accurate single measurement of position and momentum. I can, and HAVE done so. What limits me from doing it to infinite accuracy is the technology of detection, not the HUP. The HUP kicks in in my ability to PREDICT the outcome of such a measurement! That's a different question! Just because I have a plane wave and well-defined momentum, it doesn't mean any single measurement of position is undefined as if you will instead get a SMEAR all over your detector because THAT particle is supposed to be delocalized. No such thing has ever been detected. Instead, what is smeared is the position of REPEATED measurement of the identical system! When faced with such smearing of many position measurement, one will conclude that predicting where the particle is going to be is hopeless!

I have tried to explain this in painful detail in my journal entry on this. Obviously, if you have read it, I haven't done it with the clarity that I thought I did. A definite single measurement doesn't imply a definite knowledge of the system. I can make very good measurement of position and momentum - this not the HUP. However it doesn't mean that my knowledge of that can be used to predict with arbitrary accuracy the NEXT time I make the same measurement. Now THIS is the HUP!

Zz.

[Hans de Vries]

HUP should always be placed in it's historicaly context as the basis of the
Copenhagen Interpretation: http://www.aip.org/history/heisenberg/p09.htm


HUP is a derived law.

Basically from Planks Law E=hf, p=h/λ relating Energy/Momentum with
Space/Time via the Fourier Transform, and applied on the Gaussian "Bell"
curve for statistical uncertainty.

Original page of Heisenberg's derivation plus explanation:
http://www.aip.org/history/heisenberg/p08a1.htm

It was only later corrected by a factor of 2.. (\hbar \rightarrow \hbar/2)

Often metaphysical talk creeps in here, when taken outside the original
scope, like: "One can choose to measure the position exact but then one
can not meaure the momentum and visa versa: One can choose to
measure momentum exact but one can not measure position.

There is no "mind over matter control" choise. When both are measured
simultanously they may be exact to all the digits of the (digital) meter.
Only repeated measurements will reveal how far they were off from the
statistical averages. The ranges (and the center values) are always pre-
determined by the describing wave-function. There's no way to influence
that afterwards.

Even worse is: "The measurement itself disturbs the precission"
Improving thechnology will continue to make more exact and less disturbing
measurements.



Regards, Hans

[Antiphon]

In what form is the uncertainty introduced in a SINGLE measurement of the momentum?


The uncertainty is introduced as follows; when the planewave was coming
in it had a definite momentum which means you knew what the momentum
would measure as before you measured it or if you measured it 6 times, you got the same
value 6 times. But after a position measurement the momentum is now
a planewave spectrum, NOT a definite value. You can of course MEASURE
only a definite value so it's the NEXT moementum measurment that FIXES
some random definite value. But between your slit and detector there was
NO DEFINITE VALUE to the momentum. It just didn't exist as a particular number.

In what range can we expect the modified momentum to be the next time
we measure it? That depends exactly and only on how tightly the position
measurment localized the particle.


I have a very thin slit. At some time, your free particle passed through it. I say "Ah ha! At time t1, there was a particle going through position x1, and my uncertainty in the position is +- width/2!"


Ok. You now have an uncertainty in your ability to PREDICT the value of
any subsequent momentum measurement. This uncertainty is of the order
\delta p = \hbar / (2 \times slitwidth) . This uncertainty in the momentum didn't exist in my
planewave but it now exists because of the dimensions of your slit and
only after you register the particle having passed through it at time t .


But being smart, I also put a CCD screen behind the slit. I can record, to ARBITRARY ACCURACY limited by my CCD technology, where the particle hit the detector. The lateral position (in the direction perpendicular to the slit) of where the particle hit tell me the momentum in this direction.


This is now a second measurement, and yes you can record to arbitrary
accuracy where the particle hits.


The uncertainty in this momentum depends only on the uncertainty in determining where the electron hits the CCD. In fact, the higher the resolution of the CCD, the higher the accuracy with which I can determine the position the particle hits the detector. I have made measurement in which the accuracy is has high as 2 pixels on a CCD! This has zero to do with the width of the slit or the HUP. And voila, I have made a DEFINITE single measurement of position x AND momentum p_x of your particle.


As I pointed out in the earlier post, the uncertainty relation does not
address the two-measurement case. Einstein pointed this out himself
with an arrangement similar to the one you have here. A definite value
for the momentum is actualized by your sensor. The particle has an
indefinite momentum after going through your slit whereas it had a definite
mometum before the slit.


I have just done what you said cannot be done.


No, you have made two measurements. I said after a single precise position
measurement it no longer made sense to say that the particle has a definite
momtum. It will ACTUALIZE a definite mometum if and only if a momentum measurment is made, like your CCD device.


In NONE of these have I said anything about "predictability". All I care about is the question "can I make as an accurate of a position and momentum measurement of a single particle?"

The answer is :yes. This is because that question depends on the technology of detection.


- but not at the same time. For a single measurment, the answer is no.



But the question: "if I make the position measurement VERY, VERY accurate (very small slit), then can I predict with equal accuracy where there the particle will hit the CCD and thus, determine it's transverse momentum?" The answer is NO.


Correct, because of the HUP.


These two are DIFFERENT QUESTIONS under different circumstances. You cannot say I cannot make accurate single measurement of position and momentum. I can, and HAVE done so.


Let's be clear. If your slit is tiny then: You made a position measurement
at the slit and have accurate position information. But now you don't
know the momentum because THERE ISN'T a DEFINITE momentum anymore.
Then you made a moementum measurement and got an UNPREDICTIBLE but
definate result. No problem with this.


What limits me from doing it to infinite accuracy is the technology of detection, not the HUP. The HUP kicks in in my ability to PREDICT the outcome of such a measurement!


Yup, and the fact that there was no definte value for the momntum
between slit and CCD, hence the reason for the Heisenberg unpredictability.


That's a different question!


It's the only case in which HUP makes any sense here.


Just because I have a plane wave and well-defined momentum, it doesn't mean any single measurement of position is undefined as if you will instead get a SMEAR all over your detector because THAT particle is supposed to be delocalized. No such thing has ever been detected.


Of course not, because that's all wrong. You WILL get a definite position
outcome from a position measurment of a planewave. But it will cease
to be a planewave thereafter which is the whole point of the HUP.


A definite single measurement doesn't imply a definite knowledge of the system.

I can make very good measurement of position and momentum - this not the HUP.


...I've said, not at the same time becuase this is the HUP.

[ZapperZ]

This is getting mind-boggling by the minute.

Here's the starting point. A and B are 2 non commutating operators. The HUP that most people can recite is that if you measure observable A, let's say, then the greater the certainty you know about A, the LESS certain you know about B.

Now so far so good.

However, here's where things are bastardized. They then go on by saying that if one were to measure A very precisely, then B is undefined and one can't measure B with any kind of accuracy. I have seen this repeated many times, even on PF. This is what drove me to write the lengthy explanation on why this is utterly false. A single value of A and a single value of B can be measured with arbitrary accuracy that does not depend on each other's accuracy. This is an instrumentation accuracy.

Now you brought up the "at the same time" issue, which I find rather strange. The whole concept of commutation relation IS the order of measurement, that AB is not the same as BA. How and where does this imply a "simultaneous" measurement? The presence of the slit is similar to causing the wavefunction to "collapse" to a single position eigenstate. NO ONE, not even me, ever argued that the momentum part is still undermined. However, it CAN be measured and when we do, it will reveal a SINGLE value the same way we measured the "single" value for the position! When I do that, I have 2 values - position with its uncertainty, and momentum with its uncertainty.

The question is, in that single measurement, are the uncertainties associated with those two values THE uncertainties in the HUP? I say no. Now is THIS what you are disputing?

Zz.

[OlderDan]

The question is, in that single measurement, are the uncertainties associated with those two values THE uncertainties in the HUP? I say no. Now is THIS what you are disputing?

Zz.
After the brief discussion related to HUP on the other thread I was almost convinced that I had simply misinterpreted your statement about HUP being a derived consequence rather than a fundamental principle, so I followed up by reading the dialog over here to better understand your interpretation. Now I see that we really do have a fundamentally different understanding of the HUP.

As I (and I believe everyone else with whom I have ever discussed this topic) understand it, the answer to your last question above is YES. What I believe is being disputed by Antiphon, as I understand his posts, and certainly by myself is your notion that simultaneous measurements of the position and momentum of a single particle can be made to any degree of precision, limited only by detection technology. The reason you see so many instances of a different interpretation of HUP from yours is simply that a lot of people have a different understanding of what it means. So either there are a lot of us misguided folk out here, or your interpretation is incorrect.

I have to agree with Antiphon that your description of the position and momentum measurements at the slit and the screen do not constitute simultaneous measurements of position and momentum. Your argument appears to rest on the assumption that particles have a highly localized momentum that is determined when they pass through the slit, which becomes manifiest when they are are detected by the screen. Implicit in that assumption is that the particle follows a direct path from the slit to the screen, never deviating from that path by more than a slit width and forcing it to hit a localized spot on the screen.

I find this interpretation to be contrary to the postulates of QM. The wave function does not confine the particle to a specific path between the slit and the screen, or give it a nearly specific momentum. In fact, the measurements you have described do not directly measure the momentum of the particle at all. What you have described is two position measurements, one at the slit and a second one at the screen. You then deduce the momentum by assuming you know how the particle got from the first location to the second. The interpretation that Antiphon has spelled out is that the first position measurement that localizes the particle at the slit necessarily leaves that particle in a state of uncertain momentum, centered in the forward direction with momentum components equally distributed to either side. The wave function does not tell us how the particle got from the slit to the screen. It only gives the probability that when the second measurement is made, the particle will be found in any localized area of the screen.

In wave mechanics, the wave function characterizes the state of individual particles. It is not just a distribution function for an ensemble of particles. An eigenfunction of the momentum operator has infinite spatial extent. The wave function of a localized particle is a wave packet consisting of a distribution of momentum eigenfunctions. The more localized the wave packet, the wider the distribution of momenta, and the wider the distribution of momenta the faster the wave packet spreads with time. The manifestation of this in the slit experiment is that the smaller you make the slit and the farther away you place the screen, the wider the wave packet will be when it reaches the screen, resulting in a wider pattern of particle hits on the screen.

[metacristi]

Heisenberg's Uncertainty Principle (HUP)

Some claim that it should be labeled Heisenberg's Indeterminacy Principle (HIP) unfortunately for them alternative views are still valid (HIP is intepretation laden with Copenhagenism and related views/interpretations). Thus from the beginning we must make a clear distinction between the different interpretations of HUP, as of now there are two main interpretations:

1. The 'soft' version, which is compatible with all valid interpretations of QM, simply says that we cannot measure complementary parameters (such as position-momentum or energy-time) simultaneously with infinite precision. The minimal explanation of this is the 'interactionist' hypothesis, according to it this principial limitation is due to the existence of a finite quanta of action (different from 0), the perturbations which appear being unfathomable (irrespective of the technology used). No assumption about the intrinsic nature of the universe (deterministic or not) is made.

2. The 'strong' version, theory ladden (with the Copenhagen interpretation and related views), goes way further by saying that a quantum particle does not have simultaneously a precise momentum and position (this being also the explanation for our principial incapacity to know). In other words nature is inherently random.

Well in spite of the actual preference in the scientific field I find the first interpretation the most rational one (though of course the latter is also rational) based on all evidence we have and the fact that there is subdetermination at quantum level. Indeed the true nature of our universe (deterministic or not) is still an open problem, all the experiments done so far are also totally compatible with fully deterministic interpretations of QM, involving non local hidden variables. There is no experimental way, as of now at least, which to make a compelling difference between existing interpretations, the Copenhagen Interpretation (and further improvements) is not at all really superior empirically to the others. Simply some variants of copenhagenism at least, have a higher degree of coherence with GR, that is with the main body of actually accepted theories, but this is not at all a necessary sign of truth.

Some touched the problem wether HUP (at least the 'softer' version) is really a prediction of the standard formalism of QM. Well it can be argued that HUP can be deduced from the standard formalism of QM but we must be very careful here for we should interpret those results statistically (consistent with Born's interpretation of wavefunction) or Heisenberg claims that the principle holds for every singular case...this does not follow (strictly deductively) from quantum mechanics and moreover there are problems with the 'frequentist' interpretations of probability.

To expand a bit the last idea is the 'softer' version of HUP (Heisenberg's Uncertainty Principle) really an universally valid prediction of the standard formalism? As I've already undelined the usual 'frequentist' interpretation of probabilities is incompatible with the claim of Heisenberg that we have the right to extend the probabilistic meaning of HUP, as derived from standard QM, to small samples or single events. The predictions made by the standard formalism of QM referrs at statistically relevant samples, identically prepared (indistinguishable practically), thus the uncertainty relations as derived from the standard formalism of QM are valid only statistically.

To extend them at singular particles would imply to assume that the wavefunction defines the status of singular particles in the sample too, in contradiction with Born's statistical interpretation assumed initially. It is compatible with a bayesian interpretation of probabilities (vastly involved in scientific practice) indeed but in the absence of any clear interpretation of probabilities neither is there a clear answer regarding its real meaning (Bayesianism has its own problems, important ones). Unfortunately some thought experiments/'Gedanken' experiments and great coherence with other (still) accepted parts of science are not enough to back Heisenberg's strong claim that HUP holds in the case of all imaginable experiments/ thought experiments.

So that, at limit, we cannot even say that HUP (the 'softer', general definition) is a prediction of the standard formalism of QM valid for all cases (including single particles). Anyway even accepting that the 'softer' version of HUP is an universal prediction of the standard mathematical formalism (this is a fully descriptive/formal deduction not a causal deduction the only one which can give real understanding by answering the 'why HUP' question!) the 'hard' fact remain: the 'indeterminacy' is totally related to the copenhagenist, rather positivist, approach. I think it would be safer to always mention after saying that Nature is inherently indeterministic the disclaimer 'at least according to the Copenhagen Interpretation and further 'improvements', preferred currently by a majority of scientists'.

[ZapperZ]

As I (and I believe everyone else with whom I have ever discussed this topic) understand it, the answer to your last question above is YES. What I believe is being disputed by Antiphon, as I understand his posts, and certainly by myself is your notion that simultaneous measurements of the position and momentum of a single particle can be made to any degree of precision, limited only by detection technology. The reason you see so many instances of a different interpretation of HUP from yours is simply that a lot of people have a different understanding of what it means. So either there are a lot of us misguided folk out here, or your interpretation is incorrect.

I have to agree with Antiphon that your description of the position and momentum measurements at the slit and the screen do not constitute simultaneous measurements of position and momentum. Your argument appears to rest on the assumption that particles have a highly localized momentum that is determined when they pass through the slit, which becomes manifiest when they are are detected by the screen. Implicit in that assumption is that the particle follows a direct path from the slit to the screen, never deviating from that path by more than a slit width and forcing it to hit a localized spot on the screen.

And this is where I am baffled. Maybe I blanked out when I wrote "simultaneous measurement" earlier in this thread (did I ever?). However, if you read either my journal entry on the Misconception of the HUP, or my last entry regarding what really is meant by non-commutation, I don't think I ever implied making an "instantaneous" measurement of BOTH non-commuting observables!

The WHOLE issue that prompted my original essay on this in my journal was in direct response to the repeated claims that once one has made a measurement of one observable, one can no longer THEN make any accurate measurement of the other! This means that in the single-slit case, AFTER I have determined the position with very "good" certainty (particle passing through a slit), what that statement is saying is that I can no longer determine with any reasonable accuracy the momentum of the particle. This is FALSE, and I can demonstrate EXPERIMENTALLY, not just in principle, that this is false. All I need to know is where the particle hits the detector AFTER the slit. I have then determined its momentum. The accuracy to which I measure that momentum depends ENTIRELY on the accuracy of my detector. This accuracy has nothing to do with the width of the slit.

So up to this point, which part is in dispute? I have made ZERO assumption about "superposition of many momentum", etc... etc. ALL I care about is determining the position FIRST, and then determining the momentum NEXT. This is purely an act of measurement, something experimentalists like me like to do.

Now if you ASK me to tell you what the momentum of the NEXT particle that passes through the slit will be, then my ability to accurately give you the answer depends VERY MUCH on the width of the slit, i.e. the degree of certainty of the position measurement. The smaller the slit (the smaller Delta(x)), the less certain is my ability to predict where the particle will hit the detector, and thus, the larger the Delta(p_x) will be.

So again, up to this point, which part is in dispute?

Note that I HAVE done similar measurments that I have described. The electrons in the conduction bands of metals are described almost by sum of plane waves. In a photoemission experiment, the in-plane momentum of these electrons are conserved upon being photoemissted from the surface plane. This means that BEFORE a measurement, it has the same sum of various plane waves. If I use a hemispherical electron analyzer such as the Scienta SES, I can make as accurate determination of the momentum as I want, limited ONLY by the pixel size on my CCD screen! It is accepted that what is detected IS the in-planed momentum of the electron while it is in the material - there is a clear one-to-one correspondence! How do we know this? The band structure we obtained agrees with theoretical band structure of "standard metals"![1]

The accuracy of determining where one particle is, and its momentum is detector dependent. This is not the HUP. This is INSTRUMENTATION!

Zz.

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

[OlderDan]

And this is where I am baffled. Maybe I blanked out when I wrote "simultaneous measurement" earlier in this thread (did I ever?). However, if you read either my journal entry on the Misconception of the HUP, or my last entry regarding what really is meant by non-commutation, I don't think I ever implied making an "instantaneous" measurement of BOTH non-commuting observables!

The accuracy of determining where one particle is, and its momentum is detector dependent. This is not the HUP. This is INSTRUMENTATION!

Zz.


I'm truly sorry if I keep misinterpreting you, but in fact you have made statements that sound like you are making a claim for simultaneously meausring both quantities. One of them is the last paragraph in the quote above, and another is the quote that follows

But being smart, I also put a CCD screen behind the slit. I can record, to ARBITRARY ACCURACY limited by my CCD technology, where the particle hit the detector. The lateral position (in the direction perpendicular to the slit) of where the particle hit tell me the momentum in this direction. The uncertainty in this momentum depends only on the uncertainty in determining where the electron hits the CCD. In fact, the higher the resolution of the CCD, the higher the accuracy with which I can determine the position the particle hits the detector. I have made measurement in which the accuracy is has high as 2 pixels on a CCD! This has zero to do with the width of the slit or the HUP. And voila, I have made a DEFINITE single measurement of position x AND momentum p_x of your particle.

I can't see how you can make this statement, but I believe it is representative of the statements you have made that make some of us think you are talking about "simultaneous measurement" of both observables. You have not made a single measurement that determines the position AND the momentum of the particle. You have made two measurements of the postion of the particle at two different times, and from those deduced what the momentum of the particle must have been between those two measurements to get the particle from the first position to the second position.

If you had chosen the word OR instead of the word AND when making your statements about measuring both position and momentum, then I would have no problem agreeing with what you are saying.

The WHOLE issue that prompted my original essay on this in my journal was in direct response to the repeated claims that once one has made a measurement of one observable, one can no longer THEN make any accurate measurement of the other! This means that in the single-slit case, AFTER I have determined the position with very "good" certainty (particle passing through a slit), what that statement is saying is that I can no longer determine with any reasonable accuracy the momentum of the particle. This is FALSE, and I can demonstrate EXPERIMENTALLY, not just in principle, that this is false. All I need to know is where the particle hits the detector AFTER the slit. I have then determined its momentum. The accuracy to which I measure that momentum depends ENTIRELY on the accuracy of my detector. This accuracy has nothing to do with the width of the slit.
I have no disagreement with the first part of this. I agree that the claims you are attributing to others are FALSE. Measuring the position of the particle at some point by passing it through a slit demands that its wave function be a superposition of a wide range of momentum eigenfunctions, any one of which could become the momentum observed in a subsequent momentum measurement. The point that needs to be clear is that you cannot predict which of those momentum values is going to be measured before making the measurement. I agree with you that passing the particle through the slit does not preclude making a later momentum measurement with arbitrary precision.

The second part of your last paragraph however gives me pause. If measuring the position of the particle by passing it through a slit forces it into a state where its momentum is distributed over a wide range of values, how can detecting its arrival on a screen within one or two pixels be interpreted as a measurment of its momentum with high precision? The measurement that locates the particle on the screen should have exactly the same effect with regard to its momentum as the position measurement at the slit. The smaller you make the grid of your CCD to precisely locate the position of the particle, the less you know about its momentum at the time of that position measurement.

I question whether you are making momentum measurements at all in the quantum sense. A momentum measurement would require that you somehow determine the wave vector of the particle, and a relatively precise measurement of the wave vector can only be obtained if the position of the particle is relatively unknown. That would require something like a phased array that gives up precision of position in order to achieve precision in determining direction of arrival. I'm not convinced that a classical view which says that to get from one point to another in a certain amount of time the particle must have had a certain momentum during transit yields a valid measurement of the particle's momentum at the screen, or at the slit, or anywhere in between. I question your claim that you have measured the particle's momentum by determining where it hits the screen.

[ZapperZ]

I can't see how you can make this statement, but I believe it is representative of the statements you have made that make some of us think you are talking about "simultaneous measurement" of both observables. You have not made a single measurement that determines the position AND the momentum of the particle. You have made two measurements of the postion of the particle at two different times, and from those deduced what the momentum of the particle must have been between those two measurements to get the particle from the first position to the second position.

If you had chosen the word OR instead of the word AND when making your statements about measuring both position and momentum, then I would have no problem agreeing with what you are saying.

Ignoring that fact that I did not explictly mention the word "simultaneous" in that paragraph, what if I said instead "Voila! I have made a definite measurement of postion, and I have also made a definite measurement of momentum"?

There is never any "time period" in applying AB and then BA. All that implies is that one applies one AFTER the other, in sequence. I assumed that this is known. Obviously, I should have stressed that.

The second part of your last paragraph however gives me pause. If measuring the position of the particle by passing it through a slit forces it into a state where its momentum is distributed over a wide range of values, how can detecting its arrival on a screen within one or two pixels be interpreted as a measurment of its momentum with high precision? The measurement that locates the particle on the screen should have exactly the same effect with regard to its momentum as the position measurement at the slit. The smaller you make the grid of your CCD to precisely locate the position of the particle, the less you know about its momentum at the time of that position measurement.

I question whether you are making momentum measurements at all in the quantum sense. A momentum measurement would require that you somehow determine the wave vector of the particle, and a relatively precise measurement of the wave vector can only be obtained if the position of the particle is relatively unknown. That would require something like a phased array that gives up precision of position in order to achieve precision in determining direction of arrival. I'm not convinced that a classical view which says that to get from one point to another in a certain amount of time the particle must have had a certain momentum during transit yields a valid measurement of the particle's momentum at the screen, or at the slit, or anywhere in between. I question your claim that you have measured the particle's momentum by determining where it hits the screen.

At the risk of repeating what I have written in my journal, here we shall go again.

1. Slit is at postion y1 with a width of Delta(y), slit oriented along x-direction. Plane wave particles moving along z-direction impinging normal to the slit.

2. Particle passes through the slit. When this occurs, all I can say is that at that instant, the particle is at y1 location where the slit is, and my uncertainty in its position corresponds to the slit width Delta(y).

3. Particle leaves the slit, hits a detector at location L after the slit. The detector produces an image at the a y-location, call that y2.

4. If the slit is small enough, y2 can differ, and differ greatly from y1. So it has "drifted" off center along the y-direction by an amount y2-y1, i.e. it has a y-component of momentum, something it didn't have before.

5. You know the z-momentum before, and since you didn't do any position constaints along the z-direction (and x-direction), the p_z momentum remains unchanged (there can be relativistic corrections here if necesary which is way too long to describe). You then know how long it takes for the particle to travel from the detector.

6. It is then a matter of geometry and straightforward mechanics to find the y-component of velocity and momentum. I now have p_y.

The accuracy of measuring y1 and y2 are instumentation accuracy. They are not dictated by the HUP. As I know more about y1, the pixel size on my detector do not automatically becomes larger, nor is the spot size on the detector when the particle hits it blows up.

Nowhere in here did I claim to measure ALL the momentum components, only the one affected by the presence of the slit (position measurer). In the Scienta analyzer, the orientation of the slit determines in which crystallographic direction are measuring the momentum. We NEVER measure the full momentum since (i) the out-of-plane momentum is not conserved and (ii) the slit is only along one particular direction.

So again, what is in dispute here?

Zz.

[OlderDan]

6. It is then a matter of geometry and straightforward mechanics to find the y-component of velocity and momentum. I now have p_y.

The accuracy of measuring y1 and y2 are instumentation accuracy. They are not dictated by the HUP. As I know more about y1, the pixel size on my detector do not automatically becomes larger, nor is the spot size on the detector when the particle hits it blows up.

Nowhere in here did I claim to measure ALL the momentum components, only the one affected by the presence of the slit (position measurer). In the Scienta analyzer, the orientation of the slit determines in which crystallographic direction are measuring the momentum. We NEVER measure the full momentum since (i) the out-of-plane momentum is not conserved and (ii) the slit is only along one particular direction.

So again, what is in dispute here?

Zz.
The dispute is that a classical mechanical computation of a change in position divided by a change in time is being used to deduce the momentum of a quantum particle everywhere along a trajectory between two position measurements. The wave function of the particle that is squeezed through the first slit is collapsed to a localized wave packet in the y direction, but not into a y-momentum eigenstate, or even a narrow spectrum of y-momentum eitenstates. The smaller you make the slit, the less you can know about the y-momentum of the particle. The second measurement of the postion of the particle collapses the wave function a second time, but it does not determine the momentum of the particle. If it did, then if the screen had a single second slit the trajectory of the particle would continue along the extended line between the slits. That is not what QM says will happen. If there were a second slit you would have the same uncertainty in the momentum of the particle passing through the second slit that you had when it left the first slit. If there were multiple slits in the screen, the particle would not be forced to pass through any one of them and you would have a multi-slit interference distribution on another screen farther along the path. I submit that you have not measured the momentum of a single particle by capturing it on the screen. By repeating the experiment with many particles you have found the spread of the wave packet at the screen, from which you can infer the momentum distribution of the wave function of the particles that made it through the first slit. That distribution is consistent with the degree of momentum uncertainty suggested by the HUP.

[Antiphon]

OlderDan has articulated everything I was thinking but has done a much better job
of expressing it, and using the correct terminology as well. ZapperZ, I think you do
have a handle on the HUP. But there must be a subtle miscommunication between
us all about it which still leaves me feeling uneasy as I cannot spot it.

Nevertheless I think the discussion in this thread of the HUP speaks for itself for
those who are able to follow it, so I will not add to it any further. I encourage
the rest of you to go on though and I will follow the thread from a distance.

[ZapperZ]

The dispute is that a classical mechanical computation of a change in position divided by a change in time is being used to deduce the momentum of a quantum particle everywhere along a trajectory between two position measurements. The wave function of the particle that is squeezed through the first slit is collapsed to a localized wave packet in the y direction, but not into a y-momentum eigenstate, or even a narrow spectrum of y-momentum eitenstates. The smaller you make the slit, the less you can know about the y-momentum of the particle. The second measurement of the postion of the particle collapses the wave function a second time, but it does not determine the momentum of the particle. If it did, then if the screen had a single second slit the trajectory of the particle would continue along the extended line between the slits. That is not what QM says will happen. If there were a second slit you would have the same uncertainty in the momentum of the particle passing through the second slit that you had when it left the first slit. If there were multiple slits in the screen, the particle would not be forced to pass through any one of them and you would have a multi-slit interference distribution on another screen farther along the path. I submit that you have not measured the momentum of a single particle by capturing it on the screen. By repeating the experiment with many particles you have found the spread of the wave packet at the screen, from which you can infer the momentum distribution of the wave function of the particles that made it through the first slit. That distribution is consistent with the degree of momentum uncertainty suggested by the HUP.

But you somehow are ignoring the fact that I DID measure a location on the detector and should be able to deduce THAT particular momentum of THAT particle for THAT instant. I didn't say that this is the ONLY possible momentum for the NEXT particle. In fact, if I repeat this a gazillion times, I will get a spread in the measured value of momentum. This spread would correspond exactly to the HUP as dictated by the slit width!

If you have a problem with this, then you should also have a problem with Bell-type experiments. According to your position, the polarization that I would measure is simply ONE of all the possible superposition of polarizations and therefore, not the "true" polarization.

Again, I would appeal to the ALREADY established method that is used in angled-resolved photoemission spectroscopy.[1] What I had described is no different than what is done is this technique. If we are getting the "wrong" momentum, then the published results are nonsense and faulty and have nothing to do with any kind of "momentum". If you think this is so, then it is imperative that you alert the various publications on this.

Zz.

[1] Valla et al., Science v.285, p.2110 (1999).

[OlderDan]

But you somehow are ignoring the fact that I DID measure a location on the detector and should be able to deduce THAT particular momentum of THAT particle for THAT instant. I didn't say that this is the ONLY possible momentum for the NEXT particle. In fact, if I repeat this a gazillion times, I will get a spread in the measured value of momentum. This spread would correspond exactly to the HUP as dictated by the slit width!

I am not ignoring the fact that you measured the location of the particle on the detector screen. What I said, very explicitly, is that the measurements you made are both position measurements at two different moments in time and that you then did a classical calculation to deduce the momentum of the particle from those two measurements. You did not measure the momentum of the particle in either location. You assumed that because you knew where the particle was in two different locations that the particle must have taken a direct route to get from one place to the other. In post #24 in this thread you stated

1. How well I know the location of a photon depends on how wide a slit I make, no? The smaller the slit, the more I know about where that photon was when it passed by it.

2. When it passed by the slit, what is its transverse momentum perpendicular to that slit? Unless its momentum changed between the slit and the detector, then I can make the assumption that this momentum remained the same between the moment it passed through the slit and the moment it hits the detector. All I need to do is figure out WHERE on the detector it hits. Then, using simple geometry, I know it's momentum in that direction. How well I determine that momentum depends on the resolution of my detector, i.e. how well can I determine where it hits the detector. The larger the number of pixels on my CCD camera, for example, the finer I can determine this location.
I can't quite pin down what you are saying about "simultaneous" measurements of position and momentum of a siongle particle. When I bring up the word, you seem to not want it attributed to yourself as in your remark in #46, but this is what you said in #6 (the bold highlights are yours, not mine)


The HUP is NOT about the uncertainty in INSTRUMENTATION or measurement. One can easily verify this by looking at HOW we measure certain quantities. It is silly for Heisenberg to know about technological advances in the future and how much more accurate we can measure things. This is NOT what the HUP is describing. The HUP is NOT describing how well we know about the quantities in a single measurement. I can make as precise of a measurement of the position and momentum of an electron as arbitrary as I want simultaneously, limited to the technology I have on hand. I can make improvements in my accuracy of one without affecting the accuracy of measurement of the other.
The issue I have raised is whether or not the assumption of "constant" momentum from the moment the particle passes through the slit until the time it reaches the screen that you claim you can make is valid. It is not valid from the point of view a single-particle interpretation of the wave function or state variable of the particle, and that is the argument I have been presenting. If you believe that you can measure the momentum of the particle in this manner, you must reject the single-particle interpretation and replace it with an ensemble average interpretation, which seems to be the theme of your posts on this issue.

Maybe this puts you in good company. I make no pretense of being current or anywhere near the forefront of the development of thinking on these issues. Far greater minds than mine have debated the interpretation for decades, and apparently the debate continues. Maybe the tide is flowing your way. I did make some effort to review what others have said before responding here again and came across this site

http://www.phys.tue.nl/ktn/Wim/muynck.htm#quantum

I am too far removed from the front to judge where the ideas presented here fit into the mainstream of thinking these days. I have not read all of it, and if I did I would not fully grasp all of the implications, but at least it exposed me to a point of view that seems closer to yours than to what this author refers to as the "standard formalism" that is the basis for my arguments. But even here, while the problem of simultaneous measurement of "incompatible observables" (like position and momentum) under the standard formalism is viewed as being somewhat out of touch with real world "nonideal" measurements, I don't see a claim being made for unlimited precision of simultaneous measurements of two such observables.

I have a problem dismissing the single-particle interpretation in favor of a pure ensemble interpretation. It goes back to what got me involved in this discussion in the first place on the other thread. I don's see how you can relegate the HUP to a statement about the inability to predict the measurement of the momentum of the NEXT particle, as opposed to future measurements of the momentum of the particle you whose position you just measured, and still use it to make arguments abut why electrons cannot be confined to nuclei or postulate the existence of "virtual particles" as quantum fluctuations involved in the forces of interaction between material particles as other great thinkers have done.

I've done as much as I am going to do to pursue this discussion. I will leave it to guys like you who are actually doing science instead of just talking about it to sort it all out in the end.

[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 :smile: (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. o:)

[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:
http://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 doesnt come above c ! So they may exist inside the nucleus.

If this is the case,Why isnt 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 suprise 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:
http://www.physicsforums.com/showthread.php?t=83213 :


If this is the case,Why isnt 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 :biggrin: .

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 instantanous 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 instantanous 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 suprise 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 happend with momentum, I have to make many measurements. How else am I gonna 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 happend with momentum, I have to make many measurements. How else am I gonna 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 in to 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 any one 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 any one 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 any one 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 travelling 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. :uhh:

[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 beleive 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
aquire 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 any one 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 beleive 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.