"We cannot measure both position and momentum..."

  • #1
[Mentors' note: This thread's prefix has been set to 'B']
We all know that the quote in the title is an imprecise convenience when talking about the heisenberg uncertainty principle in a context where we would not want to enter into conceptual or fundamental issues to make a more correct statement. But what is the correct statement according to the originators of quantum theory? I have summarized my understanding below:

The reason we don't have to talk about measurement in classical physics is the fact that we can always control and account for the influence of the measuring bodies on the objects under investigation. For example we can make the effect of the measuring bodies as small as we want, or if it is finite, we can control and take that finite effect into account in our description. This means that we can talk about the state of a system, for example the position of a particle, as something that exists independently of observation. This is not possible in quantum physics because the effect of the measuring bodies is uncontrollable. If a body is to serve as a clock, then there will be an uncontrollable exchange of energy with the clock, which cannot be separately taken into account in order to specify the state of the objects. Any attempt to do so would interfere with the capability of the object to serve its original purpose of providing an account in time of the objects under investigation, so we are faced with the task of analysing the possibilities of definition and observation in view of this circumstance.

Any experiment where we attempt to prove that "an amount of energy E went into the clock" will destroy essential features of the original phenomenon which we are trying to specify by measuring the exchange of energy with the clock. The very functioning of a body as a measuring device is incompatible with a simultaneous control of energy and momentum exchanged with it. Thinking along classical lines, we might say that “we will not measure the energy exchanged with the clock, but can’t we still posit the existence of such a thing even in the absence of measurement?”

If you posit that a particle passes through one or the other of the slits of a double-slit experiment, then there is a logical consequence, which is that there is no interference pattern. The idea that a particle passes through one or the other of the two slits implies that the probability of arriving at a particular point is the sum of the probabilities of arriving through each of the slits, i.e. there is no interference. Since however you do observe an interference pattern in certain situations, this assumption is wrong. If you measure which slit it passes through, then you will destroy the interference. This is why you can't talk about the state of the particle as something independent of what you are experimentally doing. You simply cannot assume that the particle passes through one slit or another if you are not measuring it, because the conclusions from it will be wrong. This situation is different from the idea that there is some exact state which you are unable to measure.

Consider a diaphragm with a slit, through which a particle passes. Say we have measured the momentum of the diaphragm before the passage of the particle. Now, once the particle has passed through the slit, we are free either to repeat the momentum measurement, or to measure the position of the diaphragm. So, without disturbing the particle which has already passed through the slit, we can predict either its initial position or its momentum. Einstein concludes from this that both the initial position and its momentum must be real properties of the system. Bohr argues that the possible types of predictions regarding the future behavior of the particle depend on what you choose to do with the diaphragm, even though you are not “interfering with the particle after it has passed through the slit.” The state of a particle is not an independent property of the particle itself, but is tied up with the conditions of the experiment, so you can disturb the state without interfering with the particle by influencing the conditions of the experiment.The idea of 'state' in quantum theory ill defined without a specification of the whole experimental arrangement. Even though the particle has already passed through the slit, the meaning of 'state' is still inextricably connected to what you do to diaphragm, because that is part of the experimental procedure. Disturbing the conditions of the experiment is equivalent to a disturbance of the state, a word which cannot be applied to the second system by itself, but rather only to experimental set up as a whole. The fact that one cannot control separately or somehow take into account the effect of the measuring apparatus on the system in order to specify the state of the objects, like it was possible in classical physics, means that there is no sharp distinction between an independent 'state' of the objects and the measured interactions with the experimental setup.
 
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  • #2
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Some paragraphs would have helped.

The uncertainty principle is not about measurements. It is inherent to the objects. They simply do not have exact momenta, positions and energies in general. This has nothing to do with your measurements or the other setup.
 
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  • #3
Yes, your statement is also something that is used as a convenience but imprecise. The question is, what does it mean to "have" and why not? What is the essential difference with classical physics, and why does the existence of h change the idea of objective properties like energy and momentum, position?
 
  • #4
I would say that in fact measurement is the central issue in understanding this.
 
  • #5
Nugatory
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Yes, your statement is also something that is used as a convenience but imprecise. The question is, what does it mean to "have" and why not? What is the essential difference with classical physics, and why does the existence of h change the idea of objective properties like energy and momentum, position?
A good explanation would require going into the math, which would mean spending some quality time with a college-level QM textbook; the "Science and math textbooks" section of Physics Forums will have some good recommendations.

Without the math.... We would say that a particle has an objective momentum if we can put it into a state such that a measurement of its momentum has a 100% probability of giving us a particular value. Likewise we would say that a particle has an objective position if we can put it into a state such that a measurement of its position has a 100% probability of giving us a particular value.

However, when we go into the math of how these probabilities are calculated, we find that the state in which there is a 100% chance of measuring a particular value for the position is necessarily one in which a range of position results are possible, and vice versa.

It is tempting to interpret this as saying that the particle still has a definite position, but we don't know what it is - but it turns out that for many other observables (the qualification is here as a nod to the Bohmians, and I humbly beg them to withhold their maledictions in this thread) this interpretation predicts subtly different behavior than the "no objective value"interpretation. Where it has been possible to run experiments checking for this difference, the experiments have falsified the "definite value, but we don't know what it is" model.
 
  • #6
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I think it's correct. Heisenberg has formulated the principle just thinking of the perturbation introduced in the measure of a particle!
How do we talk about the uncertainty principle without talking about measurement? Indetermination on impulse is an indeterminacy of what?
 
  • #7
Stephen Tashi
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But what is the correct statement according to the originators of quantum theory?
You are posing a question about history. What year do you wish to use in dividing the "originators" of quantum theory versus the "subsequent developers" of the quantum theory?

It's easy to predict that the thread will digress from matters of history to the contemporary understanding of the uncertainty principle.
 
  • #8
When I said originators, I only meant to say I was interested in the "orthodox" understanding of the uncertainty principle, rather than the various interpretations. I guess I should have said, "what is the correct statement according to the copenhagen interpretation, or rather, according to bohr?"

So, no i am not interested in the history. Just the "standard" view, or "orthodox" view, rather than interpretations.
 
  • #9
Stephen Tashi
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I guess I should have said, "what is the correct statement according to the copenhagen interpretation, or rather, according to bohr?"
It's controversial whether Bohr's views represent a contemporary orthodox view. There's also controversy about how to state a contemporary version of the Copenhagen interpretation However, you have made it clear that you aren't asking about history.

Your question does have a cultural aspect. It asks about an "orthodox" view. Someone familiar with the various textbooks now in use could tell us whether there is a majority opinion.
 
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  • #10
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It's controversial whether Bohr's views represent a contemporary orthodox view.
There's also controversy about how to state a contemporary version of the Copenhagen interpretation However, you have made it clear that aren't asking about history.
Your question does have a cultural aspect. It asks about an "orthodox" view. Someone familiar with the various textbooks now in use could tell us whether there is a majority opinion.
The so-called "orthodox" interpretation "is the one defended by the Copenhagen school, Bohr in the lead.
All textbooks, fundamentally, follow this "interpretation", since it is the most logical and intelligible, in addition to the fact that it is fully confirmed by the physics and the formulas that are at the base. Some books, for "cultural" information, advise the student that other "interpretations" have been developed (which do not change an iota to mathematics) to highlight that there are "problems" to accept the "orthodox" version, as the Einstein-Bohr controversy teaches us.
 
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  • #11
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When I said originators, I only meant to say I was interested in the "orthodox" understanding of the uncertainty principle, rather than the various interpretations. I guess I should have said, "what is the correct statement according to the copenhagen interpretation, or rather, according to bohr?"
We could have a long digression about what is the "orthodox" interpretation.... But it would be a digression. The uncertainty principle is baked into the mathematical formalism of quantum mechanics - if X and Y are incompatible observables (position and momentum, for example) then the states of definite X are necessarily not states of definite Y; and this mathematical formalism applies to all interpretations.
 
  • #12
HAYAO
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I was convinced quite well by a youtube video that explains the fundamental properties of wavefunction, which is the uncertainty principle in a general sense.

Although I am not a physicist myself to be able to discuss this in detail, I am pretty sure the fact that we try to express a state with wavefunctions (or a vector in Hilbert space) directly leads to the consequence that there is a uncertainty principle. (In a more general sense, uncertainty principle applies for non-commuting operators. Position and momentum operators obviously do not commute.) In another words, I very much have to agree with mfb's post.

So the question of what that really means, seems to me like a trivial question because it is the way it is. Expressing a state with a vector in Hilbert space sounds like an axiom to me, and with that starting point, it follows that uncertainty principle holds for position and momentum measurements. But of course, correct me if I'm making errors. I'm also interested.

(slightly off-topic question: time and energy also have uncertainty relation, but how was that derived? There is no time operators, right?)
 
  • #13
atyy
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More general than the uncertainty relations are the commutation relations. One interpretation of the commutation relations is certainly that position and momentum cannot be simultaneously measured (as stated in the title of this thread).
 
  • #14
To get started, I would say we focus on Heisenberg's first paper on the uncertainty relations, and an important note added at the end of the paper. Heisenberg notices in his paper that the existence of h implies that observation always involves finite and discontinuous changes. For example, even if you use a wave of very small intensity, the light quantum idea implies that the energy arrives in finite quantities of amount E = hf. From considerations of this type, Heisenberg derives the formula δx δp ~ h.

A note added at the end of the paper, Heisenberg admits that his understanding of his relations as expressed in his paper is wrong. Bohr has explained to him that the reason for the uncertainty relations cannot be ascribed merely to discontinuous exchanges of energy alone, because our inability to eliminate experimental disturbances does not by itself imply that the quantities don't have objective values. The real reason and meaning of the uncertainty relations is that the concepts used to describe the different aspects of information obtained about say a particle have this character, so it is not that we cannot simultaneously measure, but that we cannot simultaneously define these concepts. If we observe the light emitted by a body, the energy and momentum of the photons can only be defined in terms of the frequency and wavelength of the radiation. And the only wave of attaining a connection with space-time pictures is by having a wave restricted to a finite region of space. Any wave restricted to a small region of space has a large indefiniteness in its wavelength. In this theory, what we have to realize is that E = hf is our only means of defining the concept of energy. The definition of energy and momentum in terms of spacetime pictures, i.e. p = mv etc, is only valid in the so-called geometrical optics limit of the wave theory. We disturb the classical relations in the definition, and that is the solution.
 
  • #15
DrChinese
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A note added at the end of the paper, Heisenberg admits that his understanding of his relations as expressed in his paper is wrong. Bohr has explained to him that the reason for the uncertainty relations cannot be ascribed merely to discontinuous exchanges of energy alone, because our inability to eliminate experimental disturbances does not by itself imply that the quantities don't have objective values. The real reason and meaning of the uncertainty relations is that the concepts used to describe the different aspects of information obtained about say a particle have this character, so it is not that we cannot simultaneously measure, but that we cannot simultaneously define these concepts.
There are several elements of the principle (HUP) that have not been mentioned in this thread.

a) You can simultaneously measure non-commuting attributes on entangled particles. This does NOT allow you to "beat the HUP". So apparently, the disturbance that might be attributed to a measurement is NOT a factor.
b) You can measure P(x) and Q(y) - i.e. commuting observables - on a particle to any desired degree of precision, and neither of those measurements need not prevent the other's accuracy from being maintained. So again, measurements don't really seem to be the issue.
c) You can measure some attributes repeatedly and get the same answer every time. Why is it that only non-commuting observables are affected by measurements, and not the one being re-measured?
d) And of course there is Bell's Theorem.

Put these together, and the only reasonable explanation is that "we cannot simultaneously define these concepts". This conclusion is outside of a specific interpretation, as obviously there are interpretations that are designed to get around this.
 
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  • #16
There are several elements of the principle (HUP) that have not been mentioned in this thread.

a) You can simultaneously measure non-commuting attributes on entangled particles. This does NOT allow you to "beat the HUP". So apparently, the disturbance that might be attributed to a measurement is NOT a factor.
b) You can measure P(x) and Q(y) - i.e. commuting observables - on a particle to any desired degree of precision, and neither of those measurements need not prevent the other's accuracy from being maintained. So again, measurements don't really seem to be the issue.
c) You can measure some attributes repeatedly and get the same answer every time. Why is it that only non-commuting observables are affected by measurements, and not the one being re-measured?
d) And of course there is Bell's Theorem.
About (a), can you give details? I in fact did talk about entanglement in my first post. "A particle which has been in interaction with another body for a short time cannot be said to have a state independent of the other body even after the interaction. Disturbing the conditions of the experiment has an essential influence on the 'state' of the particle which has already passed through the diaphragm" is simply Bohr's way saying that the diaphragm and the particle are entangled.

When you say non-commuting operators can be simultaneously measured for entangled particles, could you give an example?
 
  • #17
DrChinese
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About (a), can you give details? I in fact did talk about entanglement in my first post. "A particle which has been in interaction with another body for a short time cannot be said to have a state independent of the other body even after the interaction. Disturbing the conditions of the experiment has an essential influence on the 'state' of the particle which has already passed through the diaphragm" is simply Bohr's way saying that the diaphragm and the particle are entangled.

When you say non-commuting operators can be simultaneously measured for entangled particles, could you give an example?
Sure. In a typical scenario, you might have 2 entangled electrons. You measure spin in X direction on Alice, measure non-commuting spin in Y direction on Bob. But you have NOT learned anything about Bob from your measurement on Alice, as you will see NO correlation between Alice's X and a later measurement of Bob's X spin.

On the other hand: you COULD (conceptually) perform the following instead: You measure spin in X direction on Alice, measure momentum of Y on Bob. You will learn something about Bob from your measurement on Alice, as you WILL see correlation between Alice's X and a later measurement of Bob's X spin. That's because X spin and Y momentum commute.
 

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