Trying to understand Heisenberg Uncertainty Principle in a physical sense

[ThomasT]

Without HUP particles would follow the same trajectory to the level as enforced by experimental conditions.

Without HUP ... or without h? Or does either make a difference wrt individual measurements? For example, what's the theoretical limit on the accuracy of position measurements? ZapperZ seemed to indicate in his blog post that there isn't one. As did jtbell in his reply to Runner 1.

But of course there is a limit wrt the products of the statistical deltas of quantum conjugate measurements such as position and momentum.

What would the HUP be if there were no fundamental quantum of action? (Δx)(Δp) ≥ 0 ?

What is the essence of the difference between the quantum and the classical world? Is it a matter of scale? Does h define that scale?

This is a bit off topic, but I'm just curious wrt how to talk about this.

 

[Ken G]

What would the HUP be if there were no fundamental quantum of action? (Δx)(Δp) ≥ 0 ?

Yes, normally one creates a "classical world" by taking the limit that h--> 0. However, that classical world wouldn't work at all like ours, at the atomic scale, it would only work like ours at the macro scale. Indeed, there wouldn't be any macro scale, because the micro scale wouldn't work to hold it up (atoms would spiral into the nuclei, etc.). I guess such a classical world is a bunch of astronomical black holes and little else. What I don't get is, why didn't classical physicists worry about this in 1900? They probably did, but imagined it had some minor fix, not a whole new theory like QM.

What is the essence of the difference between the quantum and the classical world? Is it a matter of scale? Does h define that scale?

Yes, scale is the key, but it's not always length scale, it can be time scale, or even occupation-number scale. In essence, it is the scale of the action compared to h. When the former is >> the latter, we have the classical world, but when they are ~, we have the quantum world, whatever that is.

 

[Runner 1]

I like what Ken G wrote too. He's pointing out, I think, that the essence of the HUP isn't that position and frequency are distinct measurements that can't be obtained simultaneously. As ZapperZ pointed out (in his blog post), the HUP doesn't "prevent anyone from knowing both the position and momentum of a particle in a single measurement with arbitrary accuracy that is limited only by our technology".

What Ken G said is that the HUP is a manifestation of the wave mechanics, whereby there's "a connection between" the localization and the periodicity of a the wave.

So the physical essence of the HUP is that there's a relationship between, eg., position and momentum measurements, and, further, that that relationship is limited by h, the fundamental quantum of action.

Thus, the physical meaning of the HUP as it relates to measurements in experiments (which is what the OP was asking about), is that there's a relationship between, eg., statistical accumulations of position and momentum measurements (as with any two canonically conjugate quantum variables) that's basically defined by the inequality (Δx)(Δp) ≥ h .

So let me see if I understand what you and ZapperZ are saying correctly:

For one measurement of a particle, you can record values for position and momentum to as much accuracy as you like, but when this same experiment is repeated multiple times, the values that you measure each time will have standard deviations between measurements that satisfy (Δx)(Δp) ≥ h?

 

[Ken G]

I'd say the most straightforward way to do it is to prepare all the particles in the ensemble in the same way, and then do x measurements on half of them, and p on the other half. Then the variances in the x and the p measurements will obey the HUP regardless of your measurement precision (and you'll get equality, rather than >, if your precision is always better than the variances). If you want to do x measurements, followed by p measurements, on every particle, then your experimental precision for the x measurement will provide the more stringent application of the HUP than would the variances in the x measurements.

 

[ThomasT]

So let me see if I understand what you and ZapperZ are saying correctly: ...

First, let me say that I'm just an amateur, a sometime student of physics trying to understand this. ZapperZ is a professional condensed matter physicist, I think. jtbell has a Phd in physics. I think most of the contributors to this thread are graduate degreed. And I'm pretty sure that all of them are more knowledgeable wrt physics than I am.

For one measurement of a particle, you can record values for position and momentum to as much accuracy as you like, ...

This is my current understanding. And statements by ZapperZ (in his blog entry) and jtbell (in this thread) seem to support that view.

... but when this same experiment is repeated multiple times, the values that you measure each time will have standard deviations between measurements that satisfy (Δx)(Δp) ≥ h?

Yes, I think that's a basic form of the inequality for the HUP relationship between position and momentum measurements. But don't take my word for it. The info is out there on the internet. I first got introduced to this stuff via Heisenberg's little book "The Physical Principles of the Quantum Theory". From that it stuck with me that the HUP is essentially about the relationship between certain measurements (ie., as the accuracy of one measurement increases, then the accuracy of the other must decrease) and not essentially about the inability to measure them simultaneously. Even so, I vaguely remember some statements by Heisenberg that I found somewhat confusing.

I think I understand the main source of your confusion. It confuses me too. Some people say that you can't make simultaneous measurements of, eg., position and momentum. Others say that you can. The latest post by Ken G seems to indicate that whether you can or can't measure position and momentum simultaneously is irrelevant wrt the HUP.

You asked in an earlier post:

... why do people say "there is an inherent uncertainty even with a perfect measurement device"? What does that even mean? Since there is no such thing as a perfect measuring device, how can they know there is an inherent uncertainty?

I think this refers to the h, the fundamental quantum, wrt the HUP. That is, the product of the standard deviations (the uncertainties in the position and momentum measurements) can't be less than h.

 

[Runner 1]

Okay, thanks for all your replies everyone! It's still a little fuzzy for me, but I think I understand it a lot better than I did coming into this thread.

 

[ThomasT]

Okay, thanks for all your replies everyone! It's still a little fuzzy for me, but I think I understand it a lot better than I did coming into this thread.

Say exactly what it is that's fuzzy. Maybe someone will satisfactorily unfuzzy it. For example, I'm still fuzzy on whether or not simultaneous quantum measurements of, say, position and momentum, are possible, exactly what that means -- ie., does it refer to to a single measurement where two values are gotten, or to two distinct measurements, one for position and one for momentum done at the same time -- and exactly how it's done experimentally.

 

[Ken G]

I don't believe simultaneous measurements of p and x are possible to arbitrary precision, but I think you could build an experiment to measure them both simultaneously if the precision satisfied the HUP. To my thinking, the way you do an exact measurement (in principle) is, you identify some basis of eigenfunctions of the operator that corresponds to the measurement you intend to do, expand your initial state on those basis function, and then you quite purposefully "decohere" any crosstalk between those basis states. By that I mean, you attach a macro instrument that is explicitly designed to destroy the coherences between those very basis functions that you have in your superposition. Note that this cannot be done simultaneously for two non-commuting operators, their bases are linearly independent and we cannot decohere with respect to both bases at the same time.

However, this really refers to exactly precise measurements, which achieve complete decoherence for a given basis. The complementary observable would be a real mess, the probabilities of its eigenvalues would be spread out all over the map. But real measurements aren't like that-- instead, there is only some "bin" that we can have confidence our observable lies within, and that bin will have a finite width. So we are not achieving complete decoherence across all the basis functions (say |x> states), we are only decohering between basis states some delta x apart. This should allow us to also simultaneously decohere between |p> basis states, some delta p apart, so long as delta x * delta p > h or so.

So I think the HUP may well be interpreted as a limit on the simultaneous precision that is possible in complementary observables, and it also applies for constraining the variances when measuring both of the complementary observables, but only one per particle, to arbitrary precision over an identically prepared ensemble.

 

[zonde]

You have a lot of questions in your post ThomasT. I will try to answer what I can.

Without HUP ... or without h?

Without HUP. We use "h" for quantitative description of quantization.

Or does either make a difference wrt individual measurements? For example, what's the theoretical limit on the accuracy of position measurements? ZapperZ seemed to indicate in his blog post that there isn't one. As did jtbell in his reply to Runner 1.

I would say that limit on individual position measurement is classical. You can't make detector arbitrary small. Or slit arbitrary narrow.

But of course there is a limit wrt the products of the statistical deltas of quantum conjugate measurements such as position and momentum.

I think that question about momentum measurement poses a difficulty.
There are no measurement device that would measure momentum in single shot. Basically you use classical approach to calculate momentum from position measurements.
Should we factor out interference out of our momentum calculations?

And once we talk about momentum measurements what we say about dispersion? Is dispersion actually HUP?

What would the HUP be if there were no fundamental quantum of action? (Δx)(Δp) ≥ 0 ?

Sorry, I don't know what is "quantum of action".

What is the essence of the difference between the quantum and the classical world? Is it a matter of scale? Does h define that scale?

We don't have classical model for number of effects observed in quantum world. But if we assume that such model is possible then scale isn't really a factor.

 

[ThomasT]

@ Ken G,
Thanks, your elaborations are most helpful.

@ zonde,
Thanks also for your help.
By the way, you mentioned:

Sorry, I don't know what is "quantum of action".

The quantum of action is h. Here's relevant Wiki articles: Planck's constant and Action

 

[Runner 1]

Okay, whew! I've read all of your all's forum posts, many articles on the internet by various physicists, and my Quantum Chemistry book (McQuarrie), and I believe I have made sense of most everything. I think a lot of my (and thousands of other people's) confusion stems from ... semantics!

So I'll present the way I've thought everything out, and if there's still some things that don't seem quite right, let me know (and back it up with an explanation if you can):

• First, looking at QM from the viewpoint of the Copenhagen Interpretation, there is no such thing as position or momentum until the system is observed. Meaning, these concepts only exist at the time of measurement (kind of like how there is no such thing as a "lap" unless you're sitting down). Which means when you take a measurement of one particle, there is no such thing as "uncertainty" in that measurement. An "uncertainty" implies that there is a true value for the position or momentum of that particle, and the results you got from your measurement differ from the true value by the "uncertainty". Therefore, "uncertainty" is inherently a multi-measurement concept in the CI view.
  
  In this manner, let's say you have a lot of identically prepared systems of a particle. In half of the systems, you measure the position (which is now an extant concept since you're in the process of measuring). And you calculate the standard deviation of these measurements and call it $\sigma_x$. Then, for the other half of the systems, you measure the momentum. And you calculate the standard deviation of these measurements and call it $\sigma_p$. When you multiply these values together, you will always find that:
  $\sigma_x \sigma_p > \frac{\hbar}{2}$​
  This is Heisenberg's Uncertainty Principle in the Copenhagen Interpretation. Remember, this relation holds when measuring position and momentum of identical systems separately. When measuring the two together (simultaneously), a new relation applies:
  $\sigma_p \Delta x \geq \pi \hbar$​
  Here, $\Delta x$ refers to a region of x width in which a particle is localized ($\Delta x$ has a completely different meaning in bullet point 2).
  
  What this means experimentally is -- suppose you have a bunch of identically prepared systems. And you know for sure that your particle is in a region $\Delta x$ in each system (knowing this implies a measurement is being made). After measuring the momentum for each system, the standard deviation of all measurements of momentum will obey that relation.
• Second, we now look at QM from any of the various other interpretations that say there is such a thing as a "real" position and momentum (realism), regardless of whether we are in the process of measuring the system or not.
  
  In this case, the word "uncertainty" refers to the amount that a particle's measured position or momentum can differ from its "true" position or momentum:
  
  
  $\left|x_{measured} - x_{real}\right| < \Delta x$
  
  $\left|p_{measured} - p_{real}\right| < \Delta p$​
  
  As far as I know, it is not possible to formulate a relationship between these two quantities, $\Delta x$ and $\Delta p$, as doing so would imply that you could perform an experiment revealing more information than predicted with the equations of quantum mechanics. In other words, if there is such a thing as a "real" position and momentum, these values cannot be found according to any known theory.
  
  (Note that the standard deviations relationship of multiple measurements still applies, as from above.)
• Finally, for both of these cases there is a limit to how precisely you can measure position and momentum together. This is the "observer" effect and happens because by measuring something you are disturbing it. This is unrelated to the above mentioned Heisenberg Uncertainty Principle. I am unaware of an equation describing numerically the "observer" effect. I'm not even sure if this is a "hard" limit, because I've read about things like weak measurements that don't seem to disturb the system.

 

[ThomasT]

@ Runner 1,
I notice you're online so I'll say that I like your exposition, and it, along with posts from other members, have clarified some things for me.

Now, we just have to wait and see if any PFers who might be more knowledgeable about this stuff have any objections to what you wrote.

 

[vijayan_t]

This is the physical explanation of uncertainty

 

[PatrickPowers]

I'm trying to understand the Heisenberg Uncertainty Principle, as it relates to experimental measurements, because it's kind of confusing me. We just learned the derivations for it in my QM class -- basically it's two standard deviations multiplied together (corresponding to measurements of incompatible observables).

Before I explain what I'm asking, I want to be clear that I'm not trying to understand it from a "philosophical" angle, i.e. Copenhagen interpretation. I'm just trying to understand the aspects of QM as they relate to physical measurements.


So the usual soundbite is "The Heisenberg Uncertainty Principle says that the more accurately momentum or position is know, the less accurately the other one may be known".

Another favorite is "Even with a perfect measuring device, there is still an inherent uncertainty in knowing both the position and momentum of a particle".

These statements are meaningless to me, and they sort of gloss over a good physical explanation. I view them as cop outs -- like saying "Well, I don't really understand HUP, but I'm just going to repeat something everyone else says that sounds fancy and scientific so I still appear as if I know what I'm talking about".

Let's consider a hypothetical situation in which humans have perfected particle detectors down to Planck scale. And let's assume these detectors record data to a trillion significant digits. The detector is a big slab of some material, and when a particle hits it, it registers the particle's position on the slab, and the momentum as it strikes.

So what exactly does the operator of this detector see on their computer screen? Will it show two numbers, each with a trillion digits, or will the computers just shut off to prevent HUP from being violated? (I kid).

In other words, assuming we lived in a universe without HUP, how would the results of an actual high-resolution experiment differ from those with HUP?

Thanks if you can shed any light on this!

I will try.

Start with a particle or some energy. It doesn't just sit there, it jiggles around. So if you assume it just sits there, everything you try to deduce after that will come out wrong. After a few decades of having nothing work, scientists decided to assume that it jiggles and things started working. So it is pretty safe to believe that it really does jiggle.

So this random jiggle is the source of the uncertainty. The jiggle can't be gotten rid of. I think of this jiggle as being N-dimensional,. Reduce the jiggle in any of those dimensions and it increases in the remaining dimensions. You can't decrease it in all of those dimensions. There is a certain irreducible amount.

The connection between "certainty" of position and momentum is kind of a mathematical accident. This is only one of the ways to look at the jiggle. There are several other ways to look at it, but whatever you do it is always there.

The whole idea of "certainty" doesn't really apply, which could be the main reason that that principle is so confusing. It is kind of "if you wrongly assume that there is certainty, then you get this weird result."

 

[Appity]

Okay, whew! I've read all of your all's forum posts, many articles on the internet by various physicists, and my Quantum Chemistry book (McQuarrie), and I believe I have made sense of most everything. I think a lot of my (and thousands of other people's) confusion stems from ... semantics!

So I'll present the way I've thought everything out, and if there's still some things that don't seem quite right, let me know (and back it up with an explanation if you can):

• First, looking at QM from the viewpoint of the Copenhagen Interpretation, there is no such thing as position or momentum until the system is observed. Meaning, these concepts only exist at the time of measurement (kind of like how there is no such thing as a "lap" unless you're sitting down). Which means when you take a measurement of one particle, there is no such thing as "uncertainty" in that measurement. An "uncertainty" implies that there is a true value for the position or momentum of that particle, and the results you got from your measurement differ from the true value by the "uncertainty". Therefore, "uncertainty" is inherently a multi-measurement concept in the CI view.
  
  In this manner, let's say you have a lot of identically prepared systems of a particle. In half of the systems, you measure the position (which is now an extant concept since you're in the process of measuring). And you calculate the standard deviation of these measurements and call it $\sigma_x$. Then, for the other half of the systems, you measure the momentum. And you calculate the standard deviation of these measurements and call it $\sigma_p$. When you multiply these values together, you will always find that:
  
  
  
  $\sigma_x \sigma_p > \frac{\hbar}{2}$​
  
  
  This is Heisenberg's Uncertainty Principle in the Copenhagen Interpretation. Remember, this relation holds when measuring position and momentum of identical systems separately. When measuring the two together (simultaneously), a new relation applies:
  
  
  
  $\sigma_p \Delta x \geq \pi \hbar$​
  
  
  Here, $\Delta x$ refers to a region of x width in which a particle is localized ($\Delta x$ has a completely different meaning in bullet point 2).
  
  What this means experimentally is -- suppose you have a bunch of identically prepared systems. And you know for sure that your particle is in a region $\Delta x$ in each system (knowing this implies a measurement is being made). After measuring the momentum for each system, the standard deviation of all measurements of momentum will obey that relation.
  
  
  
• Second, we now look at QM from any of the various other interpretations that say there is such a thing as a "real" position and momentum (realism), regardless of whether we are in the process of measuring the system or not.
  
  In this case, the word "uncertainty" refers to the amount that a particle's measured position or momentum can differ from its "true" position or momentum:
  
  
  $\left|x_{measured} - x_{real}\right| < \Delta x$
  
  $\left|p_{measured} - p_{real}\right| < \Delta p$​
  
  As far as I know, it is not possible to formulate a relationship between these two quantities, $\Delta x$ and $\Delta p$, as doing so would imply that you could perform an experiment revealing more information than predicted with the equations of quantum mechanics. In other words, if there is such a thing as a "real" position and momentum, these values cannot be found according to any known theory.
  
  (Note that the standard deviations relationship of multiple measurements still applies, as from above.)
  
  
  
• Finally, for both of these cases there is a limit to how precisely you can measure position and momentum together. This is the "observer" effect and happens because by measuring something you are disturbing it. This is unrelated to the above mentioned Heisenberg Uncertainty Principle. I am unaware of an equation describing numerically the "observer" effect. I'm not even sure if this is a "hard" limit, because I've read about things like weak measurements that don't seem to disturb the system.

I really do like this explanation. I am also trying to think of HUP in the most concrete way possible, and I stumbled upon this thread and read through it. I'm wondering if you or anyone else can clarify some aspects of this explanation.

You write that:

from the viewpoint of the Copenhagen Interpretation[/B], there is no such thing as position or momentum until the system is observed. Meaning, these concepts only exist at the time of measurement (kind of like how there is no such thing as a "lap" unless you're sitting down).

This is sort of what I was thinking before coming into this tread . . . kind of like a "tree falls in the woods" thing. But I was thinking more along the lines that a quantum system is in an uncertain state until such a time where it's properties must be revealed in order for physics to happen (e.g. a measurement is taken, or a particle whizzing through space encounters another particle). What you say though implies that position and momentum don't exist until the system is disturbed (wavefunction collapse, I guess). Surely a quantum system that is about to encounter a situation where it's wavefuntion collapses has some sort of physical location and momentum.

Also, your point about position and momentum being simultaneously measured under CI . . . I guess I'm still looking for a more concrete explanation as to why position and momentum are in this give-and-take scenario (it's been a long time since QM, and I didn't really get it then, so bear with me!). I can just grasp that they are non-commuting, but not why. Does it make sense to ask what the position and momentum range of an undisturbed quantum system is? Is it really just the type of interaction that happens that constrains one variable and obfuscates the other?