Trying to understand Heisenberg Uncertainty Principle in a physical sense

[Runner 1]

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!

 

[HallsofIvy]

Here is one way of looking at it- you can't measure anything without changing the result. For example, to measure the temperature of water, you put a thermometer in it, the thermometer heats up (or cools down) to the temperature of the water- but that, of course, cause the water to cool down (or heat up) slightly. Or to measure the air pressure in your automobile tires, you put your pressure gauge on the tire and let air pass through it. Of course, that means you have let a slight amount of air out of the tire, so the pressure is no longer what you measured.

How do we measure the position of, say, an electron? Shine light on it so we can see it, of course! Well, not visible light, necessarily, because an electron is far smaller than the wave length of visible light and looking at something smaller than the wave length you just see a blur. We have to be talking about electro-magnetic waves of smaller wave length. But the energy of light is inversely proportional to the wave length. That is, as you make the wavelength smaller, the energy with which you are hitting the electron becomes greater- which changes the electron's momentum. That's why you have this trade off- the more accurately you measure the position by reducing the wavelength, the more you lose accuracy on the momentum.

 

[Runner 1]

Here is one way of looking at it- you can't measure anything without changing the result. For example, to measure the temperature of water, you put a thermometer in it, the thermometer heats up (or cools down) to the temperature of the water- but that, of course, cause the water to cool down (or heat up) slightly. Or to measure the air pressure in your automobile tires, you put your pressure gauge on the tire and let air pass through it. Of course, that means you have let a slight amount of air out of the tire, so the pressure is no longer what you measured.

How do we measure the position of, say, an electron? Shine light on it so we can see it, of course! Well, not visible light, necessarily, because an electron is far smaller than the wave length of visible light and looking at something smaller than the wave length you just see a blur. We have to be talking about electro-magnetic waves of smaller wave length. But the energy of light is inversely proportional to the wave length. That is, as you make the wavelength smaller, the energy with which you are hitting the electron becomes greater- which changes the electron's momentum. That's why you have this trade off- the more accurately you measure the position by reducing the wavelength, the more you lose accuracy on the momentum.

So essentially, you're measuring the movement of one moving Frisbee using another moving Frisbee? (Very loose analogy -- I know).

This is sort of how I envisioned it, but 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?

 

[DrChinese]

Another way to look at it is that there is no way to prepare a system in 2 non-commuting eigenstates. However, you can prepare a particle in commuting bases (such as position and spin).

I personally think it is best to think of it that a particle does not have simultaneously well defined values for non-commuting observables. Please note that tests with entangled particle pairs follow the HUP too. Can't beat the HUP.

You may be right when you say something about "we don't really understand" but the HUP really says all there is to say.

 

[Runner 1]

Another way to look at it is that there is no way to prepare a system in 2 non-commuting eigenstates. However, you can prepare a particle in commuting bases (such as position and spin).

Okay, I think this helps a lot...

Is there a citeable experiment that has actually tested HUP? Or is it too small in terms of scale such that we only have the theory at the moment? (I'm just curious to see an actual experiment directly affected by the limit so I can understand this a bit more).

 

[DrChinese]

Okay, I think this helps a lot...

Is there a citeable experiment that has actually tested HUP? Or is it too small in terms of scale such that we only have the theory at the moment? (I'm just curious to see an actual experiment directly affected by the limit so I can understand this a bit more).

Here is an example, you can google and find more:

http://www.news.cornell.edu/stories/sept06/schwab.quantum.html

 

[xts]

Is there a citeable experiment that has actually tested HUP?

You may make such experiment yourself in half an hour at no cost.

Take a piece of alufoil, make (with a pin) a tiny hole in it, illuminate it with toy laser and watch the Airy's pattern on the opposite wall. Now - make a bit bigger hole and watch how the pattern changes.

That is exactly the experiment you want: as you measure precisely the position (small hole) the pattern is wide (transverse momentum of the photons gets uncertain, so the direction gets also uncertain). As you measure the position with poor precision (large hole) - the pattern gets narrow, as the momentum uncertainity is much smaller now.

As you know the wavelength (thus longitudinal momentum of photons), size of the hole and angular size of the pattern (thus ratio of transverse to longitudinal momentum) - you may check they are consistent with Heisenberg's principle.

 

[jtbell]

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

Yes.

But if you fire another identically-prepared particle at it (from some kind of gun that is built as precisely as possible), you get a different set of two numbers. Fire another particle, you get yet another different set of two numbers. After you've fired a lot of particles and recorded the results, you calculate the standard deviations of both the position and the momentum. These are \Delta x and \Delta p, and they satisfy the HUP.

Furthermore, from detailed knowledge of the particle gun's construction, you can (in principle) predict only statistical quantities relating to the particles' positions and momenta at the detector, such as the mean (a.k.a. expectation value) and standard deviation.

 

[Runner 1]

Yes.

But if you fire another identically-prepared particle at it (from some kind of gun that is built as precisely as possible), you get a different set of two numbers. Fire another particle, you get yet another different set of two numbers. After you've fired a lot of particles and recorded the results, you calculate the standard deviations of both the position and the momentum. These are \Delta x and \Delta p, and they satisfy the HUP.

Furthermore, from detailed knowledge of the particle gun's construction, you can (in principle) predict only statistical quantities relating to the particles' positions and momenta at the detector, such as the mean (a.k.a. expectation value) and standard deviation.

Huh... not what I was expecting. So you actually do get numbers, but in a sense these numbers aren't "correct" (meaning you can't pin the nature of the particle down to two numbers) because they vary each time an identical experiment is performed?

 

[DrChinese]

Huh... not what I was expecting. So you actually do get numbers, but in a sense these numbers aren't "correct" (meaning you can't pin the nature of the particle down to two numbers) because they vary each time an identical experiment is performed?

You can prepare particles in identical states. Their non-commuting bases will respect the HUP. With any individual particle, you can get numbers, but they don't really mean anything if they cannot be used to predict the result of another experiment.

The classical example might be a red striped sock. Next time I look at it, it is red AND striped. Quantum particles don't work that way. And yet they do have attributes that remain the same UNTIL you look at the other non-commuting attribute.

1. So for a classical particle: Red, red. Striped, striped. Red, red. Striped, striped. I.e. it is always red and striped.

2. So for a quantum particle: Red, red. Striped, striped. Green, green. Striped. Green. Plaid. I.e. what is it at any time? Who knows? And yet it will remain plaid until you check something else. Of course this is the ideal case, in practice you may disturb it additionally and get slightly different results. This is what can cause additional confusion.

 

[Runner 1]

You can prepare particles in identical states. Their non-commuting bases will respect the HUP. With any individual particle, you can get numbers, but they don't really mean anything if they cannot be used to predict the result of another experiment.

Okay. Your posts are really helpful.

I have one more question. What you are saying is that identically prepared experiments give different results, and that it is only the statistics of these results that can be calculated. Is this correct?

And if so, is it possible that the state of a particle is dependent on the time elapsed since some universal reference time? In other words, the experiments are identical in all respects, except that they are performed at different points in time. (Of course I know this isn't possible -- this sort of question is simply to understand why not).

 

[1mmorta1]

The machine you describe will only measure position. The result will be an extremely precise measurement of WHERE the particle WAS. HOWEVER, the data will be useless, because taking such a precise reading on position will make the velocity infinitely immeasurable, and you will have no way of even pondering where the particle has ended up.

 

[1mmorta1]

Runner1, your thinking too much of particles as, well, particles. They are not. And they are not wave functions either...they are a duality. All the time. It is the choice of the observer to perform experiments which demonstrate one property or the other. The really tricky thing is that the wave/particles seem to be transcendent in that, whichever property you demonstrate, going back in time will reveal that property continuosly for the wave/particles tested, as if they knew which experiment you were going to perform. Very difficult to grasp conceptually, but proven mathematically.

 

[Runner 1]

Runner1, your thinking too much of particles as, well, particles. They are not. And they are not wave functions either...they are a duality. All the time. It is the choice of the observer to perform experiments which demonstrate one property or the other. The really tricky thing is that the wave/particles seem to be transcendent in that, whichever property you demonstrate, going back in time will reveal that property continuosly for the wave/particles tested, as if they knew which experiment you were going to perform. Very difficult to grasp conceptually, but proven mathematically.

I'm not thinking of them as anything really. My question is about turning vague statements such as "They are not [particles]. And they are not wave functions either...they are a duality" into something experimentally demonstrable. And yes, for that particular sentence, the double-slit experiment demonstrates the wave-particle duality. My question concerns an analogous experiment for the Uncertainty Principle ...which DrChinese provided a link to, so my question is mostly resolved.

EDIT: Thought I'd explain with a table for you:

Vague statement:         Electrons are waves & particles       An electron\'s position & momentum are inherently uncertain
Clarifying experiment:   Double-slit experiment                ?

 

[DrChinese]

And if so, is it possible that the state of a particle is dependent on the time elapsed since some universal reference time? In other words, the experiments are identical in all respects, except that they are performed at different points in time. (Of course I know this isn't possible -- this sort of question is simply to understand why not).

Actually, a particle in an eigenstate remains in that state until something changes it. So basically, no, time is not the factor.

For example: imagine you have 2 clone particles (this is feasible). You can measure ANY idenctical observable on these and get the same result - always. And yet... if you measure different observables, say X and Y (whatever those happen to be, but they are non-commuting), you will find that neither happens to have Y and X as their paired value. Instead, the difference will be consistent with the HUP.

 

[haael]

All problems with understanding Uncertainty Principle arise from thinking of particles as tiny hard balls. It is sufficient to imagine them as clouds to grasp the idea of fundamentally uncertain position.

To get why position and momentum measurements are incompatible, we need to employ a bit more maths. First, we must accept that "position" and "momentum" are only macroscopic concepts and do not need to actually exist.
Imagine any function. Now try to check its support (domain subset where it is nonzero) and period. It's easy to see that not all functions have any period (only periodic functions for that matter) and support is often bigger than a single point.

Actually, assigning a quantum particle position and momentum is just like trying to describe function's support and period with just one number. You cannot have both at the same time. Periodic functions have infinite support, functions with narrow support are not periodic.

Particles are not tiny balls. They are more complex concept.

 

[Runner 1]

All problems with understanding Uncertainty Principle arise from thinking of particles as tiny hard balls. It is sufficient to imagine them as clouds to grasp the idea of fundamentally uncertain position.

To get why position and momentum measurements are incompatible, we need to employ a bit more maths. First, we must accept that "position" and "momentum" are only macroscopic concepts and do not need to actually exist.
Imagine any function. Now try to check its support (domain subset where it is nonzero) and period. It's easy to see that not all functions have any period (only periodic functions for that matter) and support is often bigger than a single point.

Actually, assigning a quantum particle position and momentum is just like trying to describe function's support and period with just one number. You cannot have both at the same time. Periodic functions have infinite support, functions with narrow support are not periodic.

Particles are not tiny balls. They are more complex concept.

I think you replied to the wrong thread. But good explanation for particle/wave duality!

 

[1mmorta1]

Ah I see what you were asking. Do you feel satisfied with the answer you have to the question you were asking?

 

[Runner 1]

I'm just trying to understand the aspects of QM as they relate to physical measurements.

...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?

In addition to the experiment DrChinese provided, I also found one on my own that is a GREAT explanation to the question I asked:

http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensation"
For Bose-Einstein condensate, Wikipedia says:

In the image accompanying this article, the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: since the atoms are trapped in a particular region of space, their velocity distribution necessarily possesses a certain minimum width. This width is given by the curvature of the magnetic trapping potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution.

So in a world without HUP, the peak would be infinitely narrow. With HUP, it forms a mound. I think this example should be mentioned in QM classes as a way that something as abstract as the Uncertainty Principle relates to real data. I believe it would clarify a lot of things for many students, much like the double-slit experiment does for wave-particle duality.

 

[ThomasT]

Hi Runner 1, I'm just curious why you didn't expect the reply that jtbell gave in post #8. You said in the OP that you wanted to understand aspects of QM (such as the uncertainty relations) as they relate to physical measurements. The basic inequality (Δx)(Δp) ≥ h seems to me to communicate a pretty clear physical meaning.

 

[Ken G]

So in a world without HUP, the peak would be infinitely narrow. With HUP, it forms a mound. I think this example should be mentioned in QM classes as a way that something as abstract as the Uncertainty Principle relates to real data. I believe it would clarify a lot of things for many students, much like the double-slit experiment does for wave-particle duality.

Indeed, the HUP is another ramification of wave-particle duality, so these are the same effect. Have you heard of "Fourier transforms"? One may view these as the explanation of the HUP, as soon as one accepts wave-particle duality, because the "HUP" for a wave is that to get a waveform that is confined into a region delta x, it requires a superposition of plane waves (which have infinite extent) with a wavenumber width delta k (akin to momentum in QM) that obeys delta x times delta k ~ 1. This also shows why h shows up in the HUP-- h is the fundamental connection between k and p, via deBroglie's famous p = hk/2pi. So the bottom line is, you always get some form of HUP any time you say that particle dynamics are ruled by wavelike quantities (here the "wave function").

 

[jewbinson]

The beginning of http://www.youtube.com/watch?v=KokditqpAJg" is a great illustration of Heisenberg's uncertainty principle at work.

If you want to measure the momentum of an electron RIGHT NOW, then you look at it, and the very action of looking at it changes it's momentum BECAUSE, when you look at an object, a photon is emitted from the electron into your eye. Just like a rocket changes it's momentum when it releases it's fuel compartment.

 

[Runner 1]

Hi Runner 1, I'm just curious why you didn't expect the reply that jtbell's gave in post #8. You said in the OP that you wanted to understand aspects of QM (such as the uncertainty relations) as they relate to physical measurements. The basic inequality (Δx)(Δp) ≥ h seems to me to communicate a pretty clear physical meaning.

The basic inequality says that you can't measure two properties. My question was, why not? What's stopping you from doing that? If I gave you an equation saying you can't lift up an apple, you would say that's silly, and go lift up an apple to prove me wrong. Similarly, if someone tries to go prove the HUP wrong experimentally, what happens?

But everyone answered that for me -- I'm just clarifying what my question was.

 

[ThomasT]

The basic inequality says that you can't measure two properties.

Interesting. I see it as specifying a quantitative relationship between the measurements of the two properties.

EDIT: I think I should have phrased my reply differently. Something like: But the two properties can be measured. And the HUP specifies a quantitative relationship between those measurements, as limited by h, the fundamental quantum of action.

 

[Runner 1]

Interesting. I see it as specifying a quantitative relationship between the measurements of the two properties.

Hmm. That makes a lot more sense. Then why does everyone say it the way I just did?

EDIT: On thinking about it some more, the way you have phrased it makes a LOT of sense to me!

 

[ThomasT]

Hmm. That makes a lot more sense. Then why does everyone say it the way I just did?

I don't know. And maybe I'm wrong in how I think about it (and will need to be corrected by someone more knowledgeable than I am). But jtbell's reply (post #8) to you seemed to me to be the most straightforwardly correct answer to your question about the physical meaning of the uncertainty relations.

And then there's the mathematical understanding of the HUP which involves things which I'm assuming you haven't gotten into in depth yet.

 

[Chronos]

I like KenG's explanation. The HUP is a mathematical fact of nature. The non-commuting part Ken spoke of is what prevents you from measuring both position and momentum simultaneously, not the consequence of perturbations introduced by the measurement process.

 

[jfy4]

I have always found this entry very helpful.

http://physicsandphysicists.blogspot.com/2006/11/misconception-of-heisenberg-uncertainty.html"

 

[ThomasT]

... what prevents you from measuring both position and momentum simultaneously ...

This phrasing is what a lot of people find confusing. As Zapperz states in his blog on the HUP (linked to by jfy4 in post #28):

I have shown that there's nothing to 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.

I agree with your statement that the HUP is a "mathematical fact of nature", insofar as the canonical commutation relation is a mathematical fact and h (the fundamental quantum) is a fact of nature.

The physical meaning of the HUP as it relates to measurements in experiments, which is what the OP was asking about, is that the product of the statistical spreads about the two related measurements can't be less than the value of h -- which is due to the formulation of QM and which is not, as you say, just "the consequence of perturbations introduced by the measurement process".

 

[Mordred]

https://www.physicsforums.com/showthread.php?t=534819
Hermitian operators

 

[Mordred]

Indeed, the HUP is another ramification of wave-particle duality, so these are the same effect. Have you heard of "Fourier transforms"? One may view these as the explanation of the HUP, as soon as one accepts wave-particle duality, because the "HUP" for a wave is that to get a waveform that is confined into a region delta x, it requires a superposition of plane waves (which have infinite extent) with a wavenumber width delta k (akin to momentum in QM) that obeys delta x times delta k ~ 1. This also shows why h shows up in the HUP-- h is the fundamental connection between k and p, via deBroglie's famous p = hk/2pi. So the bottom line is, you always get some form of HUP any time you say that particle dynamics are ruled by wavelike quantities (here the "wave function").

Ok now I'm confused again (no surprise lol) In another thread below I referred to HUP in regards to a statement that one cannot visulaize both a particle and a wave at the same time, I'm pretty sure I read that somewhere but can't recall where. However it was pointed out to me that HUP utilizes Hermitian operators rather than other complementary parameters.

In light of that and the post does HUP apply to the statement I made, with KenG informative post? I am having some difficulty distinquishing the two. Thread mentioned is below

https://www.physicsforums.com/showthread.php?t=534819

 

[Ken G]

The way to think of it that works for me is to just say that what the object "is" is a particle, and that is how it manifests itself when detected. But where the particle "goes" is described by wave mechanics. So the spatial extent of the wave amplitude tells you where the particle will be found, and the undulations in the wave amplitude tells you how that "where" is changing with time (i.e., its momentum). For a particle with mass, the wave is highly dispersive, so its group velocity depends sensitively on its wavelength. In the limit of very small wavelengths, allowing very localized "wheres", classical physicists simply mistook this for the concept of a "trajectory", so they missed the wave mechanics that was there all along. The HUP is a manifestation of that wave mechanics, a connection between the concept of "how localized" and "how periodic" is the wave-- with the tradeoff that high localization requires sacrificing strict periodicity, and vice versa. There isn't any need to refer directly to "perturbing the system" in measurement, it suffices to buy off on the wave mechanics as telling the particle where to go and where to be found.

 

[Chronos]

The position of a particle is distinct from its frequency [momentum], and it's impossible to define its position and frequency simultaneously - IOW, a dumbed down version of what Ken said.

 

[zonde]

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?

Without HUP particles would follow the same trajectory to the level as enforced by experimental conditions.
With HUP particles simply don't do that below certain limit no matter what conditions you enforce on them.

I think this as well quite clearly shows difference between classical world and quantum world.

 

[ThomasT]

The position of a particle is distinct from its frequency [momentum], and it's impossible to define its position and frequency simultaneously - IOW, a dumbed down version of what Ken said.

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 .

 

[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?