Heisenberg imcertainty principle (get it)

[mrfeathers]

I don't 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? I am not trying to disprove it or anything, so don't 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 a lot 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 until 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 until 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 statement 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 don't 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? I am not trying to disprove it or anything, so don't 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&lt; (\Delta p_x)^2 \right&gt; \left&lt; (\Delta x)^2 \right&gt; \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!

 

[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 ?

 

[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 = &lt;A^2&gt; - &lt;A&gt;^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 traveled 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 traveled 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!

 

[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, anyone 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 anyone 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 anyone 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.