Understanding the Uncertainty Principle

In summary, the Uncertainty Principle states that it is impossible to know with absolute certainty the momentum and position of a particle. However, it is possible to know these values "complementary" to each other- that is, they cannot both be known simultaneously.
  • #1
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Hi.

This semester I am taking a Philosophy class (introductory) and a Modern Physics class (introductory) and we just recently (in the latter) began officially learning about Heisenberg's Uncertainty Principle.

From the lecture (and other reading I've done), I am seeing that the math for determining the momentum and position of a particle are directly linked and, together, have a "lower bound". This is my understanding of the premise for the Uncertainty Principle.

Given that, I wanted to ask something to make sure I understand it properly. The principle is saying that it is impossible for us to know, fully, the momentum and position of a particle. However, and here is my question, the particle has a specific momentum and position, it is simply unknowable. Is this correct, or am I misunderstanding? Does the fact that we cannot know both its position and momentum imply it has no specific position and momentum?

Thanks, I hope you understand my inquiry.
 
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  • #2
sgoodrow said:
The principle is saying that it is impossible for us to know, fully, the momentum and position of a particle. However, and here is my question, the particle has a specific momentum and position, it is simply unknowable. Is this correct, or am I misunderstanding?

The problem starts with imagining the electron as a "billiard ball' with a definite position and a definite momentum, i.e., the classical picture. In quantum mechanics, the electron cannot be strictly imaged this way. For example, if you throw a rock into a lake, what is the position and the momentum of the waves generated?
 
  • #3
I don't follow nor think that is an applicable analogy (though I could be missing your point).

The electron (like all, complex or simple, objects) falls under the "wave-particle duality" concept, but that does not mean they are strictly waves--it also has the particle aspect. This includes position and mass/momentum, no?

It seems our equations lead us to the fact that we cannot know these two values together, but does that imply they do not exist?

This questioning was alluded to by a discussion (in my Philosophy class) on metaphysics and the question of whether anything exists outside of our minds. Does the fact that we cannot know anything outside of our minds imply it does not exist? No, of course not, maybe it does, maybe it doesn't--it is impossible to prove from within our minds. Well, extending that to this principle, does the fact that we cannot know the (precise) position and momentum of an electron imply it does not have a position and momentum?
 
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  • #4
sgoodrow said:
I don't follow nor think that is an applicable analogy (though I could be missing your point).

Here's a nice website with additional info:
http://www.colorado.edu/physics/2000/quantumzone/debroglie.html [Broken]

sgoodrow said:
The electron (like all, complex or simple, objects) falls under the "wave-particle duality" concept, but that does not mean they are strictly waves--it also has the particle aspect. This includes position and mass/momentum, no?

Depending on the circumstances, electrons exhibit either wave-like or particle-like behavior. These "like" behaviors in no way force electrons to strictly conform to the corresponding classical concepts, though.

sgoodrow said:
It seems our equations lead us to the fact that we cannot know these two values together, but does that imply they do not exist?

Why would momentum and position, understood in their classical incarnations, be applicable "as is" in a quantum sense? If the electron is a quantum beast why would you think it should behave in such a way as to offer both classical position and momentum to observers? Note that in the realm of quantum mechanics, position and momentum are complementary observables. This property doesn't exist in classical mechanics.

sgoodrow said:
This questioning was alluded to by a discussion (in my Philosophy class) on metaphysics and the question of whether anything exists outside of our minds.

Physics has nothing to say about that question, except that objective reality is assumed to exist. Otherwise, it wouldn't make much sense to do Science.
 
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  • #5
Your question could be interpreted in two different ways; one would be if you are asking whether the outcome of a measurement is decided "in advance", i.e. is there something "hidden" in the wave function which causes a measurement to come out the way it does, or is the outcome inherently unknowable? The other would be if you're asking whether the particle in question is actually "there" and behaving as a classical particle before we measure it.

The answer to the latter, as nanobug said, is definitely no, and we know this because quanta show properties that make no sense in a classical frame, such as interference and non-locality. As for the former, you would probably be better served by asking in your philosophy class. There are many interpretations of QM which give different answers to this question, and though some may be more popular than others, anyone who tells you that one has more evidence behind it than another is misinformed.
 
  • #6
It is very important to realize that a particle can not be described by a position and momentum that is given in advance, but which we do not know.

This is the crux of the argument given by Bell about local hidden variable theories. It has been experimentally proven that reality does not work that way. Specifically - one can not always describe a local (or independent) state for a particle which has "hidden" (predetermined, but unknown) values for position and momentum.

This argument relies on entanglement, which hopefully you will cover shortly in your course... but in it does lie the answer to your question - the position and momentum can not simultaneously "really" exist. Kyuzo mentioned that there are many interpretations of Q.M. but they all must agree on this point.

Whether anything exists outside our minds is a question I can not answer... :)
 
  • #7
While I appreciate your responses, I am not fully satisfied so please bear with me. ;)

As the reading you suggested mentions, we shouldn't look at the electron's orbit around an atom as a circular orbit but rather as a standing wave. But if we do this, as you seemed to suggest, what does the Uncertainty Principle mean? Does it not specifically refer to the electron's momentum and position? What does that mean if the electron is a wave? The principle does not seem to apply to the wave-like nature of the electron; only to the particle-like nature of the electron, no?

I think I have isolated my concern:

The Heisenberg Uncertainty Principle states:
"Locating a particle in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the position uncertain."

The question was:
"Does the fact that we cannot know both the specific position and specific momentum of a particle imply it has no specific position and specific momentum?"

If I am not mistaken, you answered this by saying:
"It is not a particle, it is a particle-wave, so it need not be confined to position and momentum."

But how can that be so, and if that is so, what does the Uncertainty Principle mean?
 
  • #8
sgoodrow said:
The question was:
"Does the fact that we cannot know both the specific position and specific momentum of a particle imply it has no specific position and specific momentum?"

If I am not mistaken, you answered this by saying:
"It is not a particle, it is a particle-wave, so it need not be confined to position and momentum."

Well my response was different... when one looks a bit deeper (at entire systems that are made up of more than one particle) then your question is answered:
"Yes, that is right, even if you think of it as a sort of particle (not a wave), it is incorrect to say that it has some specific position and momentum that we simply can not know (but might exist)"

I should explain my qualifications to that statement, just to be precise. Bohmian mechanics (thought up by David Bohm) is an interpretation of Q.M. where we envisage just what you say - particles are flying around with specific positions and velocities but with some probability distribution. These particles don't behave like normal particles (i.e. follow Newton's laws), but move in ways to create interference and other quantum effects.

So, to throw some extra confusion, if you have one particle in your universe, then it can be possible to interpret it as a particle flying about with some unknown, but none-the-less defined, momentum and position. As everyone begins to learn Q.M. by starting with just one particle, this can be confusing and contradicts what you get taught!

But the universe isn't made up of one particle - it's made of lots which are interacting and getting entangled with each other all the time. Bell, and the experimenters, have shown that such an interpretation is inconsistent with reality - you can not always describe particles in that way (specifically, if it is entangled). Thus, we have to conclude that the position and momentum of a particle do not exist.

sgoodrow said:
But how can that be so, and if that is so, what does the Uncertainty Principle mean?

Well, it means exactly what it says. It simply says how well we can approximately know positions and momenta of particles. It says that big objects can have relatively well defined momenta and positions, and that the everyday world appears to behave in the way our intuition works, but that our intuition is actually fundamentally wrong, especially when we look at small things. I believe you should interpret it as saying that the particle does not have a specific position and momentum.
 
  • #9
Hmm, okay, thanks for your clarification andyferris. I understand your point a lot more now, though haven't quite made up my mind as to what I think is true (need to think about it more, and get a bunch more information--especially on that entanglement stuff!).

That said, could you provide any (ideally popularized) experiments which show the particle's non-existence of specific position and momentum? That isn't to say the inability to measure their specific position/momentum, but the actual non-existence of them as you explained.

The type of experiments I am thinking of is the interaction between two particles (like in a particle collider). From what I know, we shoot a particle at another and they can interact in a particle-like manner. Whether or not we can measure the precise momentum and position of those particles isn't important (to this discussion), but rather the fact that they do or do not collide at a specific position with a specific momentum.

That is to say: if we took that event, that specific instant in time when the two particles collide, and considered it again given the same preconditions of the entire system (be that the room, planet, or universe), that the two constituent particles would interact in precisely the same way.
 
  • #10
sgoodrow said:
That said, could you provide any (ideally popularized) experiments which show the particle's non-existence of specific position and momentum?

In the double slit experiment, when an electron goes trough both slits at the same time, what is the position of the electron?

 
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  • #11
But in that experiment the entire interaction becomes rather wave-like. Isn't the interaction in a particle collider rather particle-like? How do we explain the interaction between two particles in a particle collider? If they are hitting, doesn't that mean they have a specific position and momentum?
 
  • #12
sgoodrow said:
But in that experiment the entire interaction becomes rather wave-like.

Sure, but given that we are talking about electrons, what is the position? Or why is it that we cannot easily talk about position in this case?

sgoodrow said:
If they are hitting, doesn't that mean they have a specific position and momentum?

They are not "hitting". They are interacting (exchanging energy/momentum) via photons.
 
  • #13
the way we got taught it is that the electron has a wavefunction which can give the position and a K space interpretation that gives the momentum using the de brogile idea of wave particle duality. These two properties the wavefunction and the K(wave number) function are interlinked by the Fourier transformations. We usually think of an electron as a wave packet a series of waves superimposed. If we take the modulus of the wave function and square it then we will get an area in whcih we should expect to find the position of the electron(This is a postulate as far as I am aware) this is linked by Fourier transform to the K function which gives us an expected value for the momentum.

If we have an infinite wavefunction ie we know the precise momentum of the particle/electron but know nothing of the position.

All these ideas as to how the world works eg physics are just that ideas. They are there to explain what we see. Whether they are real in an external/physical world is up to your belief. I am an idealist so I would say that they can't be shown to be real therefore they can't be real but at the same time you have to concede the possibility of them been real.
Either way it would seem that the idea of probability is built into quantum mechanics no matter wether you believe in the physical world or not.

This post would be very much nicer with some nice diagrams and stuff but I am sure you can find them about the net.

Alex
 
  • #14
The question is not whether (A & B) exists but what can be known about (A & B).
 
  • #15
nanobug said:
Physics has nothing to say about that question, except that objective reality is assumed to exist. Otherwise, it wouldn't make much sense to do Science.

This is an interesting position.

I'm all with you on that physics, like Bohr said, is about what we can say about nature, not what nature us is.

IMO, this does not imply an objective absolute reality. In fact I can't figure out how we can DEFINE an objective reality? The best definition is the collective one, ie, the agreement among a local set of subjective reality views, as communicated between them.

In the everyday sense of science, it is supposedly objective - ie results must be repeatable by others etc. However this has more do to with the science in the sociologicla context. I personally think taking some kind of objectivity idea to it's extreme is misleading.

All I can speak of is MY subjective view of reality. I communicate with others and we compare our views, and most of the time it makses sense. But I can not make the conclusion that there is some absolute objective reality out there because of that. The locally objective reality lies in our communicated, partial, agreements. But this is subject to ongoing change IMO.

This seems to be somewhat "loosely" (all analogies are flawed) analogous to Einsteins search for absolute reference frames where his conclusion was that it doesn't exists, and instead the only reference we have is the local gravitational field.

What the correspondence to the local gravitational field of the analogy in this case? :)

/Fredrik
 
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  • #16
Ah, sorry for the delay, I was getting emailed every time there was a response and as soon as I observed this phenomenon (of being emailed when there is a response) it seemed to change and I stopped getting the emails. How appropriate, eh? Anyway...
nanobug said:
They are not "hitting". They are interacting (exchanging energy/momentum) via photons.
Could you explain this a bit more? I don't have much knowledge in the area so I can't say that I understand that phenomenon of exchanging energy through photons when there were no photons present...?

The question is not whether (A & B) exists but what can be known about (A & B).
What do you mean? I understand that there could be limitations on what can be known about A & B, but I don't understand how that implies A & B don't exist.
 
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  • #17
sgoodrow said:
What do you mean? I understand that there could be limitations on what can be known about A & B, but I don't understand how that implies A & B don't exist.

I think the point is that when we are posing questions, care should be taken so that the question is a proper question from a scientific and physical viewpoint. Sometimes the difficulty in finding consistent answers to a question, is because the question is invalid in the first place. Any sequence of words that grammatically looks like a question is not necessarily a scientifically or physically well defined question.

Now that may be an interesting point. What does it mean for a question to be invalid? IMO, it means that this question can not be constructed from the premises. The formulation of the question itself, requires some additional structure, and this missing structure is also why it's not possible to find intrinsic answers. That's how I see it.

In this sense, the question "what is A & B" is simply much more poorly defined than is "what do we know about A & B".

From the philosophical view, QM puts large emphasis on this. The idea of measurable things, is the idea that proper questions are those that have measurable answers. In QM the idea at least, is that the process of answering is the measuring process. And the physically possible measurements does constraint the possible proper questions.

/Fredrik
 
  • #18
I see what you are saying but can't help but disagree. That isn't to say I take the converse stance, which would be that something can exist without being measurable, but rather that I am skeptical about either conclusion.

You stated that:
Fra said:
...this missing structure is also why it's not possible to find intrinsic answers.
And while I know it is your opinion, I see no reasoning for that deduction. Why should a 'proper question' be confined to physically measurable answers? Or conversely, why shouldn't it? Do you see my point? It is a matter of being certain and being skeptical; while indeed it is useful to make a decision, I don't feel it is scientific to neglect the other side of the argument should you be forced to make a decision (be it for experimental or theoretical reasons). Maintaining awareness of the initial argument seems imperative to scientific thought, and this seems to be something many of us are forgetting as we delve deeper into attempting to understand Q.M.

This seems to be the root of my issue with the Uncertainty Principle, and more specifically with the responses I've received. I feel as if many of you feel/know there is some reason to deduce what you said (that which I quoted), yet have not shown any. Of course, I will gladly consider my position given any reasonable explanation for that dedution. ;)
 
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  • #19
Of course what you quoted of me, is not a "deduction", so you are free to it my opinon if you like. And even if it was a deduction, it would be a subjective one.

I can not prove anything of what I said as in "formal proof". Internally my own reasoning is justified by subjective plausabilities. I don't think formal deductions are applicable to nature, except as idealisations.

There is another complexity here. From the human point of view, the question does make sense, because you obviously posed it. So it's real. But then the human brain is supposedly more complex than a particle. So the question is what the particle level constraints shaves off the set of posable questions.

I think we can agree that a lot is subjective here. And that's part of the problem as I see it, but this is also what IMHO at least makes the subjectivity plausible. The subjectivity acknowledges the subjective view - ie what I KNOW about something. And from the subjective viewpoint, what's beyond what I know is very foggy.

Maybe you'd ask: so which subjective view is "right"? Then I think that's the whole point. All views are equally right. There is no contradiction in this. Because the contradiction would appear only when you actually compare the views. And IMO, the comparasion is a physical process. And I think the supposed contradictions (which really arent' contradictions) is partyl what gives rise to non-trivial interactions. The fact that we don't see random interactions could be because there are in a sense local objectivity, as defined by the consensus of the local neighbourhood of interacting observers.

Again, this is no deduction... I'm just adding some subjective reflections. The plausability of this may be subjective too.

That said, some of what I said suggest that QM as it stands still has problems. So I didn't mean to say I have all answers. I sure don't :)

/Fredrik
 
  • #20
Very wise words fra and I think I share a simmilar belief as you that we cannot objectively know something without introducing some kind of belief/probability.

What is an objective reproduceable experiment?

One in which the observers on average agree on the outcome of the experiment. With probability in QM doesn't this make an objective result harder to achieve. Obviously in this case we must look at the probability diustrubution in order to test our theories. But QM treads sticky philosophical ground IMO.

Alex
 
  • #21
Hi! The uncertaintly principle states that we cannot measure precisely the position and the momentum of a particle at the same time.

That does not mean that the particle has no precise momentum and position, it just means that we cannot know them.

It is like the cat in the box. Is it dead or alive? Unless we measure, as observers, we do not know the outcome, nevertheless the cat in principle is in one of the two states.

Once we perform a measurement on the wavefunction of the cat, then it will collapse and give us an answer.

In Heisenberg's Uncertaintly principle once we measure position, then the wavefunction will collapse and change state. Thus a subsequent measurement cannot be on the same state. Thus we cannot know precisely the momentum and position of the particle at the same time.
 
  • #22
Gigi said:
The uncertaintly principle states that we cannot measure precisely the position and the momentum of a particle at the same time.

That does not mean that the particle has no precise momentum and position, it just means that we cannot know them.
See, now this is what I was getting at. Upon inspection of the principle I had the same understanding and judgment, yet unless I am misinterpreting the words of the others in this thread, many are at a disagreement with this statement.

Am I correct in understanding your (those who I perceive as disagreers) disagreement, or have I misunderstood your words? I do feel that up until Gigi's post a well-defined answer had not been put forth. Is it my understanding that many believe there cannot be a well-defined answer? And if so, on what basis do you make that conjecture?

Fra, you mentioned subjectivity and I would like to respond:

Be it external or internal, it remains within the confounds of reasonably logical thought; does that alone make it scientific? No; but it does make it rational, and I think many of us agree that you can apply the rules of formal argument to rational ideas, though perhaps not as strictly, and perhaps it does not yield a precise conclusion, but it does have value in the context of a discussion.

That said, your internal subjectivity should be rationalized, as you are a rational being and would not agree to something irrational; yet I ask: since it is just as well to not make a decision, why do you specifically make one? I don't mean to question the validity of your ideas, but I do mean to address the certainty about it. Though of course, I could be misinterpreting that certainty, as it is quite difficult to communicate to another human being (in person, let alone on a message board), but I feel this is a problem of post-modernity in Science and is something we should not turn a blind eye to.
 
  • #23
Gigi said:
That does not mean that the particle has no precise momentum and position, it just means that we cannot know them.

Not true! In the case of the electron that goes through both slits, what is the position of the electron? In this case, you cannot even in principle speak of the position of the electron because the thing lives at two different places at the same time!

Gigi said:
It is like the cat in the box. Is it dead or alive? Unless we measure, as observers, we do not know the outcome, nevertheless the cat in principle is in one of the two states.

Also not true. Before one measures, it is quite possible for the 'cat' to be both dead and alive at the same time.
 
  • #24
sgoodrow said:
Ah, sorry for the delay, I was getting emailed every time there was a response and as soon as I observed this phenomenon (of being emailed when there is a response) it seemed to change and I stopped getting the emails. How appropriate, eh?

If I am not mistaken, you will stop receiving emails if you don't act on them, i.e., click on the link provided.

sgoodrow said:
Could you explain this a bit more? I don't have much knowledge in the area so I can't say that I understand that phenomenon of exchanging energy through photons when there were no photons present...?

Electrons create electromagnetic fields. Electrons also respond (feel a force) when exposed to electromagnetic fields. So where do photons come from? A photon is a particle. This particle called photon is the minimum amount of energy that the electromagnetic field is willing to exchange and is called a quantum of energy (Einstein won the Nobel prize because of this; see photovoltaic effect). So, when an electron interacts with the electromagnetic field, the electron is interacting with a bunch of other particles called photons (which are nothing more than the quanta of the electromagnetic field). So, what is an electron? An electron is the quantum of a (certain) leptonic field (electrons are leptons).

Conclusion: particles are quanta of fields.
 
  • #25
sgoodrow, I'm with you on this one. I see no reason to make decisions about things we cannot know, nor assume that we will never be able to know. But there is in a fact experiments with Bell's inequality that according to most have ruled out the Bohm interpretation and other hidden variables theories. So if you haven't already, check it out.
 
  • #26
Hydr0matic said:
I see no reason to make decisions about things we cannot know, nor assume that we will never be able to know.

Are you related to Rumsfeld, by any chance? :smile:

As we know,
There are known knowns.
There are things we know we know.
We also know
There are known unknowns.
That is to say
We know there are some things
We do not know.
But there are also unknown unknowns,
The ones we don't know
We don't know.

—Feb. 12, 2002, Department of Defense news briefing

http://www.slate.com/id/2081042/
 
  • #27
However, and here is my question, the particle has a specific momentum and position, it is simply unknowable. Is this correct, or am I misunderstanding?
The notions don't apply "down there" the same way they apply "up here".

The notion of a definite momentum/position doesn't even make sense.
Someone correct me if I'm wrong :)

It is like the cat in the box. Is it dead or alive? Unless we measure, as observers, we do not know the outcome, nevertheless the cat in principle is in one of the two states.
Don't be biased. Why disqualify the cat as an observer?:)
 
  • #28
SF said:
Why disqualify the cat as an observer?:)

Because the cat is only capable of meaningfully observing one outcome,
but never the other. :-(
 
  • #29
because we cannot observe a particles state without affecting it in some way one could subscribe to realism in the form of precise position and momentum being theoretical entities or adopt constructive emperisism by noting that we cannot report values unmeasured.

the truths of the quantum world may only exist in time intervals nature prohibits probing, leaving only averages. measurements scope systems, never an element in isolation.

bell and legget may entertain you
 
  • #30
dipstik said:
because we cannot observe a particles state without affecting it in some way one could subscribe to realism in the form of precise position and momentum being theoretical entities

Because an electron sometimes behaves as a classical wave, what would its "theoretical" position be?
 
  • #31
strangerep said:
Because the cat is only capable of meaningfully observing one outcome,
but never the other. :-(

The dead cat, being a macroscopic system, is quite able to "observe" the outcome. Awareness is unnecessary. What matters is that decoherence happens.
 
  • #32
I suggest first that Gigi has it right.

The "thing", as you put it, cannot, under any circumstances, be in two places at the same time.That's why we use probability -- maybe it's here, maybe it's there. But we don't know until we measure. And by the nature of the measurement, we'll always find one electron in one place at one time; never in two or more places.

And, what are the defining characteristics of a cat that is both alive and dead? How is such a state possible?

And, how in the world can a dead cat observe? Please tell us how.

nanobug said:
Not true! In the case of the electron that goes through both
slits, what is the position of the electron? In this case, you cannot even in principle speak of the position of the electron because the thing lives at two different places at the same time!Also not true. Before one measures, it is quite possible for the 'cat' to be both dead and alive at the same time.

Regards,
Reilly Atkinson
 
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  • #33
reilly said:
I suggest first that Gigi has it right.

The "thing", as you put it, cannot, under any circumstances, be in two places at the same time.That's why we use probability -- maybe it's here, maybe it's there. But we don't know until we measure. And by the nature of the measurement, we'll always find one electron in one place at one time; never in two or more places.

And, what are the defining characteristics of a cat that is both alive and dead? How is such a state possible?

But it is possible. That was what the Delft/Stony Brook experiments were all about. If not, one can never get the coherence gap that was due to the supercurrents in those SQUIDs going in both directions at the same time.

The Schrodinger Cat states being demonstrated in those experiments (and in chemistry where you get bonding-antibonding states) are precisely the result of superpostion where by the existence of those orthorgonal states simultaneously can be detected and become a physical property of the system being manifested via a non-commuting, non-contextual observable. It demonstrates that the existence of those states at the same time isn't simply a matter of our knowledge of it. It truly is the physical state of that system.

Zz.
 
  • #34
At this point in the discussion I just wanted to make a few comments, as I think the way it is moving is precisely how I wanted it to.

First, thank you all for responding, your efforts to help me understand this distinction (though I can't say I do understand it fully just yet) are appreciated and I have learned a lot through this thread. That said, it would be wrong for me to say that this thread was not designed for an ulterior motive, namely to demonstrate that there is a rather significant amount of confusion about this topic, and this stems from poor dispersal of information, poor lectures and explanations, or poorly demonstrated evidence. I do think it is something we as a scientific community need to be as aware of as we can, as I think it is a rather fundamental problem with the direction of physics.
 
  • #35
sgoodrow said:
At this point in the discussion I just wanted to make a few comments, as I think the way it is moving is precisely how I wanted it to.

First, thank you all for responding, your efforts to help me understand this distinction (though I can't say I do understand it fully just yet) are appreciated and I have learned a lot through this thread. That said, it would be wrong for me to say that this thread was not designed for an ulterior motive, namely to demonstrate that there is a rather significant amount of confusion about this topic, and this stems from poor dispersal of information, poor lectures and explanations, or poorly demonstrated evidence. I do think it is something we as a scientific community need to be as aware of as we can, as I think it is a rather fundamental problem with the direction of physics.

Actually, I disagree.

The uncertainty principle is very well defined. If not, you cannot write a mathematical expression to it.

It is when people forget to look at it is when it gets "confusing". The fact that it involves a "statistical" nature of a set of measurement, not just one measurement, implies two very important concepts:

1. the spread in values of an observable and the corresponding spread in values of the non-commuting observable;

2. the ability to predict the next set of measurement, which reflects our knowledge of the system based on #1.

One can measure, as accurately as technologically possible, the single position and single momentum value of a particle. I had already illustrated this in the single-slit example in previous threads. The uncertainty in such one-measurement has nothing to do with the HUP. It is when one tries to make repeated measurements, or when one tries to predict the outcome of the NEXT measurement, is when the HUP kicks in. This is the one aspect of the HUP that is often misunderstood.

I'm not sure how this thread somehow got directed into the issue of superposition.

Zz.
 
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