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 [itex]\Delta x[/itex] and [itex]\Delta p[/itex], 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 [itex]\Delta x[/itex] and [itex]\Delta p[/itex], 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 .