The Uncertainty Principle

[swansont]

Wave-particle duality is really just something we use to frighten children and undergraduates.

 

I really like that.

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I wouldn’t phrase it like that myself. Particles cannot be said to “really” be a wave. That contradicts the wave-particle duality.

 

The wave-particle duality is introduced to wean physicists from classical notions — we get to hang onto the concept that there are still particles. But we say particle when we mean localized and quantized behavior. Just because you've stopped looking for the wave behavior doesn't mean it's not there.

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Isn't this a forum for scientific debate, where we are encouraged to maintain the standards of debate? Maybe the moderators can answer that one.

 

Yes, it is. One is expected to support arguments, and limit the debate to the science. If one believes a statement to be incorrect, one needs to address why they believe it to be so.

[proton]

Just because you've stopped looking for the wave behavior doesn't mean it's not there.

Who said it wasn't there? Not I. I said that wave nature and quantum nature are not the same thing. I also said that when a quantity is measured in discreetly quantized systems the result is precisely determined. What part of that are you suggesting is wrong?

[swansont]

Who said it wasn't there? Not I. I said that wave nature and quantum nature are not the same thing. I also said that when a quantity is measured in discreetly quantized systems the result is precisely determined. What part of that are you suggesting is wrong?

 

You talked about contradicting the wave-particle duality. It's what I quoted. That's what I was responding to.

[GutZ]

Thanks Severian you are the man. You the man. The whole particle thing was throwing me off...not that I totally understood everything you said but that's exactly what I need to hear.

 

I am all visual when it comes to thoughts. It's all pictures and videos so the particle/ wave thing was really starting to bug me because it seems like it was two different properties separate from each other.

 

So when it has energy that is localized it referred to as a particle. When it's diplays wave like properties is the energy distributed amongst the wave?

 

I am getting this?...If I could visualize this I would be to problem...so frustrating converting back and forth.

[Severian]

I am glad you liked it

 

So when it has energy that is localized it referred to as a particle. When it's diplays wave like properties is the energy distributed amongst the wave?

 

We usually call it a "particle" anyway, since the subject is "particle physics" but that is just semantics. In fact, if you look at a Feynman diagram, the external legs corresponding to the "particles" are actually momentum eigenstates. Since they have a well defined momentum, they are plain waves in (position) space and are spread out with no definite position. But we still (somewhat misleadingly) call them particles. (So Feynman diagrams are most definitely not depicting billiard ball like collisions, despite appearances.)

 

I am getting this?...If I could visualize this I would be to problem...so frustrating converting back and forth.

 

I think so. We sometimes get so wrapped up in elevating quantum mechanics to some weird mystical status, when really most of the stuff we have seen before in different contexts. If you think of these things as waves you will have a much better intuitive understanding. (The hard bit to grasp is wavefunction collapse into eigenstates.)

[GutZ]

Yeah clearly i need to read up on it more..."eigenstates" I dont even know what that is so...but thanks for clearly that up at least.

[Severian]

An eigenstate is just a state with a definite value of the measured quantity. So a position eigenstate is where the 'particle' is in a well defined position, while a momentum eigenstate is when the particle has a well defined momentum. The HUP is basically quantified in the fact that the particle cannot be a position eigenstate and a momentum eigenstate simultaneously. (Eigen is German for 'distinct' (as in separate), so eigenstate is literally a distinct state.)

[GutZ]

Do waves really go on forever? So you need a wave to be in a specific eigenstate to get a precise value? (or is it more in a eigenstate when you can get a precise value) A wave a can either be localized or plain wave, and It just so happens that one state gives you precise position and the other momentum, and because you can't measure something that is in both states at once we are limited to know either one or the other precisely?

 

So its more an intrinsic property of the wave and not so much the instrument, that would make more sense as HUP is theorical principle, not just a limitation of technology.

 

Closer? lol

 

This peaked wave can be decomposed into a sum of lots and lots of plain waves, each of which has a well defined momentum. This is called a Fourier decomposition.

 

Why would it be a sum of alot of plain waves? is that because waves cancel out? Is this theoretic waves or is a peaked wave actually a sum of mulitple plain waves?

 

Oh is because waves or particles (and it's a particle because it's localized) are actual in all possible states at once? As in multiple plain waves? oh God wait that superposition....

 

I think I am jumping all over the place. I think I am just going to have to take a physics course or two.

Edited July 3, 2009 by GutZ

[Severian]

Do waves really go on forever?

 

Theoretically plain waves do (that is, a wave with a precisely determined momentum). Though of course, in reality, we never see these since we can never precisely determine the momentum.

 

So you need a wave to be in a specific eigenstate to get a precise value? (or is it more in a eigenstate when you can get a precise value)

 

That would be true for a classical wave. In the quantum case, you don't need the wave to be in a momentum eigenstate before you make the measurement. But when you make the measurement, you 'collapse' the wave into an eigenstate (this is the weird but of QM). So after the measurement it is in an eigenstate. It is as if nature randomly chooses the value of momentum you will get and then fixes the wave so that that it matches the momentum you measured. If you make a second measurement, you will get the same answer, since now it is in an eigenstate. (That last statement is really only true if you make a perfect measurement the first time - otherwise the wave stays contaminated with other eigenstates.)

 

A wave a can either be localized or plain wave, and It just so happens that one state gives you precise position and the other momentum, and because you can't measure something that is in both states at once we are limited to know either one or the other precisely?

 

So its more an intrinsic property of the wave and not so much the instrument, that would make more sense as HUP is theorical principle, not just a limitation of technology.

 

Exactly

 

Why would it be a sum of alot of plain waves? is that because waves cancel out? Is this theoretic waves or is a peaked wave actually a sum of mulitple plain waves?

 

Yes, this is just a property of waves. Mathematically it is a Fourier Series. You can write any fundtion as a sum over plain waves (sines and cosines), so similarly you can construct any shape of wave out of plain waves.

[proton]

Do waves really go on forever?

Depends on the precise definition of that term. E.g. a sine wave is infinite in extent. In practice it's finite. For example; for a particle in a box the wave is non-zero inside the box and zero outside. If a particle is represented by a Gausian wave then it's large near its peak and drops off as one gets further from the peak, but mathematical never equals zero but is close enough to zero for any practical purpose.

So you need a wave to be in a specific eigenstate to get a precise value?

No. You need the state to be an eigenstate to be sure of the value you get. Any measurement yields a value whose precision depends only one the experimental set up and instrumentation used. The precision is independant on the nature of the wave.

A wave a can either be localized or plain wave, and It just so happens that one state gives you precise position and the other momentum, and because you can't measure something that is in both states at once we are limited to know either one or the other precisely?

Yes.

Why would it be a sum of alot of plain waves?

An arbitrary state is not a simple one. But any arbitrary state can be represented as a superposition of eigenstates (e.g. plain waves). Sound is like that. Any sound can be reduced to a superposition of sines and cosines.

[Baby Astronaut]

However, the Uncertainly Principle is that electrons (or any sub-atomic particles) cannot have a definite momentum AND position.

But can't an electron have a definite variable position? A moving target, if you will, can still be determined where it'll appear down the road.

 

I know once you measure or pinpoint, it changes the thing being measured.

 

But what if there's a way around it? Look at my questions to Severian for an idea of this, below.

 

 

The wave has infinite extent, never dieing off to zero, and looks the same everywhere (apart from the up-down motion).

...the 'wave' is a very peaked object around the particle's position (and dies off rapidly as we move away from that position.

1. Ok, by "dies off rapidly" and having infinite extent, can I describe the wave as follows?

 

You have the one part where all its energy is bunched up, the "particle" aspect of it. From that particle area, the rest of the wave quickly diminishes in magnitude/concentration, like so: by 1/2, then 1/2, again 1/2, until infinity (an oversimplified and mathematically inaccurate example, but I'm just looking for the gist of it).

 

 

2. If that's how it works, does the wave stretch in two directions only (as usually portrayed)? Or infinite directions (like a blast radius going in all directions?)

 

 

3. Now, if the wave's magnitude indeed tapers off quickly, why not attempt to measure it a good distance from its "particle" area? There it might be weak enough that if one measured a good distance away from the concentration of energy -- on both sides of it -- the results might be used to deduce the position/momentum of the energy between without disturbing the particle area.

 

 

4. So everything is a wave. Then, can I accurately illustrate the whole Earth as a bunched collection of those "particle areas" held close by gravity, and waves taper off infinitely from each particle (it'd resemble an image of Earth and wisps stretch and taper off from the planet). I'd like to try drawing an oversimplified example.

 

 

5. And on waves tapering off endlessly...isn't there a rule in Physics to avoid infinities (like a singularity)?

 

 

This peaked wave can be decomposed into a sum of lots and lots of plain waves, each of which has a well defined momentum. This is called a Fourier decomposition. Since each of these plain waves has a different momentum, we can't use any one to define the momentum of the particle. It doesn't have a definite momentum because it is made up of lots of different momentum plain waves. And the more peaked the distribution becomes, the larger the range of momentum we need in our plan waves.

So a wave is really a collection of smaller waves. A wave of waves, so to speak.

 

And regardless if the wave's main thrust is going in one direction, the individual plain waves are going in their own directions within it.

 

Or so I gather.

 

 

Everything is a (quantum) wave, period.

So are gauge bosons waves? Or a form of energy? Or both/neither?

 

The only sense in which we have point-like particles...

Since you're so good at explaining the other stuff, what is a point-like particle, exactly? Couldn't quite grasp it from Wikipedia and online sources.

 

 

When a measurement is made the state falls into one if its eigenstates...

Does that mean (using the example by Severian of plain waves going in various directions), the larger wave crest or "particle" -- once measured-- would then (consequently) newly adopt the position and momentum of one of its underlying plain waves?

 

(Others can answer too )

 

 

You can't because our theories are built on Quantum Field Theory. Notice the word 'field' here. A field is an object with a value for each point in the space, just like a wave. These quantum fields are indeed waves.

Fields is another term I couldn't really grasp. But after this discussion, I might have a better idea.

 

If wrong, please correct me.

 

A field would be every part of a wave? From its particle area to everything tapering off from it? So if one traced the wave's shape, the result is the field.

 

If so, I visualize it as having a bulky middle, and progressively getting thinner.

 

A shape maybe a bit like the side view of a galaxy (if waves move in two directions only)

 

 

Would a field be shaped like that galaxy above (yet more infinitely stretched)?

 

 

Thanks Severian you are the man. You the man. The whole particle thing was throwing me off...not that I totally understood everything you said but that's exactly what I need to hear.

Second that.

 

I am all visual when it comes to thoughts.

The same reason it helped me understand. If really complicated or not intuitive, gotta be able to visualize it, then I'll grasp it a lot better.

 

 

An eigenstate is just a state with a definite value of the measured quantity. So a position eigenstate is where the 'particle' is in a well defined position, while a momentum eigenstate is when the particle has a well defined momentum.

Would position eigenstate be analogous to coordinates (for example, GPS)?

[GutZ]

I'd kiss you guys...good thing this is a forum!

 

*dances with glee*

 

I still need to do some research and take a few math classes, but I am not hopeless it understand it seems.

 

joy.

[Royston]

So after the measurement it is in an eigenstate. It is as if nature randomly chooses the value of momentum you will get and then fixes the wave so that that it matches the momentum you measured.

 

This has (since studying physics) bugged me, it's a stochastic process. If I assumed an ideal system where I could get a series of measurements, i.e a number of eigenstates over a period of time, I could trace back a history of those events, so they appear determined. However, the next event I wish to measure, to add to this series, is subject to probability.

 

It's bugged me, because it sticks the act of measurement, in a position of central importance, but the laws of physics play out without observation. Which, I thought recently, that the first paragraph, is more a mathematical consequence, over an observed phenomena. But I'm not sure.

[Xittenn]

Now we need to know a bit about the properties of waves. A wave with a single precise momentum is what we call a plain wave and looks like the traditional sine wave that you see all over the place. The wave has infinite extent, never dieing off to zero, and looks the same everywhere (apart from the up-down motion). So we can't say where this wave is at all - it is everywhere. This is one part of the HUP - if the wave has a well defined momentum, it is spread out over all space.

 

Is this why the electromagnetic plain wave is mathematically represented using a logarithmic function?

 

[math] exp[2 \pi (\frac{u \cdot r}{\lambda}- \nu t)] [/math]

Edited July 4, 2009 by buttacup

[swansont]

Plane wave. It's a plane wave, as in the wavefront is a plane.

 

 

Is this why the electromagnetic plain wave is mathematically represented using a logarithmic function?

 

[math] exp[2 \pi (\frac{u \cdot r}{\lambda}- \nu t)] [/math]

 

You're missing an "i" in the exponential, which turns it into a complex sinusoidal function

[Severian]

But can't an electron have a definite variable position? A moving target, if you will, can still be determined where it'll appear down the road.

 

If you perfectly measure its position, then it will have an uncertain momentum, so you won't know where it is oing. If you perfectly measure its momentum, you will have no idea where it is.

 

You can still predict things statistically though

 

1. Ok, by "dies off rapidly" and having infinite extent, can I describe the wave as follows?

 

You have the one part where all its energy is bunched up, the "particle" aspect of it. From that particle area, the rest of the wave quickly diminishes in magnitude/concentration, like so: by 1/2, then 1/2, again 1/2, until infinity (an oversimplified and mathematically inaccurate example, but I'm just looking for the gist of it).

 

Yes, more or less.

 

2. If that's how it works, does the wave stretch in two directions only (as usually portrayed)? Or infinite directions (like a blast radius going in all directions?)

 

I am not sure what you mean by this. We live in 3D space (4D if you include time), so I presume you mean 3 dimensions? The wave exists in 3 space dimensions.

 

3. Now, if the wave's magnitude indeed tapers off quickly, why not attempt to measure it a good distance from its "particle" area? There it might be weak enough that if one measured a good distance away from the concentration of energy -- on both sides of it -- the results might be used to deduce the position/momentum of the energy between without disturbing the particle area.

 

That is part of the "weird" bit. QM wavefunction collapse is non-local. It doesn't matter what you do, you will collapse the wavefuntion everywhere. (Though in this case, you would probably get a null result, so only collapse the wavefunction into a superposition of states which exclude the place that you measured, so probably the wavefunction won't change that much.)

 

4. So everything is a wave. Then, can I accurately illustrate the whole Earth as a bunched collection of those "particle areas" held close by gravity, and waves taper off infinitely from each particle (it'd resemble an image of Earth and wisps stretch and taper off from the planet). I'd like to try drawing an oversimplified example.

 

I was with you until the 'wisps'. For large macroscopic objects, the combined wavefunctions because statistical averages, so they behave like classical, predictable objects again. Let me use and analogy: I have no idea when you are going to die, but if I look at a large sample of the population, I can very accurately predict when they will die on average. I can then make 'macroscopic' predictions like how much it would cost to give them a pension. Similarly the Earth consists of rather a lot of particles, so although I can't predict the movement of any individual particles, I can predict the motion of the ensemble very well. This is why classical physics can emarge from quantum physics.

 

5. And on waves tapering off endlessly...isn't there a rule in Physics to avoid infinities (like a singularity)?

 

Again, I am not sure what you are getting at.

 

So a wave is really a collection of smaller waves. A wave of waves, so to speak.

 

Sort of, though no choice of 'basis' wave (ie. in your mind, the form of the waves that make up the others) is better (or more fundamental) than any other choice. We talk about plane waves in position space because they are momentum eigenstates, but there are plenty of other functions (types of wave if you like) which form complete sets. In the QM description of the Hydrogen atom for example, it is useful to describe things in terms of Laguerre functions.

 

So are gauge bosons waves? Or a form of energy? Or both/neither?

 

They are also waves (though we usualy call them fields). The photon is a gauge boson, and is also called an electromagnetic wave.

 

Since you're so good at explaining the other stuff, what is a point-like particle, exactly? Couldn't quite grasp it from Wikipedia and online sources.

 

A point particle is simply a particle which we haven't detected any substructure for. Or in other words, if we measure its position, we find it is in one place and only one place (to the accuracy of our measurements). There is not one bit over here and another over there.

 

We used to think that protons were point particles, but then we found out that they were made up of quarks and gluons inside.

 

Does that mean (using the example by Severian of plain waves going in various directions), the larger wave crest or "particle" -- once measured-- would then (consequently) newly adopt the position and momentum of one of its underlying plain waves?

 

Yes, exactly. If you measure the particle's momentum (exactly) then it becomes the plane wave associated with that momentum.

 

Fields is another term I couldn't really grasp. But after this discussion, I might have a better idea.

 

If wrong, please correct me.

 

A field would be every part of a wave? From its particle area to everything tapering off from it? So if one traced the wave's shape, the result is the field.

 

A field is just a number for every point in space (and usually time). For example, the temperature in a room is a field because it it has a value (21 degrees here, a little hotter by the radiator) for every point in the room. A wave is similarly represented by a number for each point in space (for example, a wave on the ocean might be represented as the height of the water above average sea level for every point on the ocean).

 

Would position eigenstate be analogous to coordinates (for example, GPS)?

 

No - the coordinate is the value that you measure (actually called the eigenvalue). The eigenstate is the wave that you particle collapses into once that measurement is made. Every eigenstate has an eigenvalue.

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Plane wave. It's a plane wave, as in the wavefront is a plane.

 

Sorry - that was my fault. I think I systematically wrote 'plain' earlier.

Edited July 4, 2009 by Severian
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[Xittenn]

 

Sorry - that was my fault. I think I systematically wrote 'plain' earlier.

 

I second that.............

[swansont]

 

Sorry - that was my fault. I think I systematically wrote 'plain' earlier.

 

And it collapsed the wave function in the thread.

[Baby Astronaut]

Thanks, Severian. I understand half of it a lot better.

 

Now to elaborate on a few points, and things you didn't catch what I meant.

 

 

does the wave stretch in two directions only (as usually portrayed)? Or infinite directions (like a blast radius going in all directions?)

I am not sure what you mean by this. We live in 3D space (4D if you include time), so I presume you mean 3 dimensions? The wave exists in 3 space dimensions.

 

 

The way I'm visualizing the particle waves, each one stretches horizontally to infinity, so that it resembles the ocean waves above.

 

But the height of each is limited: there's no infinite height.

 

So each ocean wave above is 3-dimensional too, but it only stretches "infinte" along one dimension: horizontally. And its length is just as limited on the frontward-rearward axis.

 

If particles are similar, I'd visualize them like the waves above, except off-camera their height dwindles rapidly...until infinity (but only in the horizontal directions).

 

Yet even if the particle/wave resembled those caused dropping a stone into a pond (examples below), they would still have a limited height.

 

 

The only alternate scenario I envision is for the particle/wave to have infinite height, length, and width.

 

 

Now' date=' if the wave's magnitude indeed tapers off quickly, why not attempt to measure it a good distance from its "particle" area? There it might be weak enough that if one measured a good distance away from the concentration of energy -- on both sides of it -- the results might be used to deduce the position/momentum of the energy between without disturbing the particle area.[/quote']That is part of the "weird" bit. QM wavefunction collapse is non-local. It doesn't matter what you do, you will collapse the wavefuntion everywhere[/b']. (Though in this case, you would probably get a null result, so only collapse the wavefunction into a superposition of states which exclude the place that you measured, so probably the wavefunction won't change that much.)

I don't see it as weird, but there's a part that still doesn't fit.

 

But, only if the particle wave's infinity occupies more than 1 dimension.

 

However...if the particle wave stretches infinitely on 3 dimensions, and if tampering with any part of it makes its wavefunction immediately collapse, then here's the problem: the wave exists in all parts of the universe at once (being infinite), and so it's always being disturbed -- even by us just waving our hands midair, anywhere.

 

Thus, if disturbing even the most remote part of the wave collapses it, then how is it any non-collapsed waves exist at all? They'd be disturbed by practically all of existence interacting with it.

 

 

So everything is a wave. Then' date=' can I accurately illustrate the whole Earth as a bunched collection of those "particle areas" held close by gravity, and waves taper off infinitely from each particle (it'd resemble an image of Earth and wisps stretch and taper off from the planet). I'd like to try drawing an oversimplified example.[/quote'']I was with you until the 'wisps'.

I just meant that since each particle is really an infinite-length wave, I can therefore just illustrate the *bunched* section of those waves (the actual particles) as residing on Earth, with the infinite parts of each tapering off into space -- and which quickly dwindle away -- so that each wave going off to infinitity is seen (by the viewer) like a barely noticeable wisp that quickly fades into non-detection.

 

 

And on waves tapering off endlessly...isn't there a rule in Physics to avoid infinities (like a singularity)? [/quote']Again' date=' I am not sure what you are getting at.[/quote']

Well people have mentioned that infinities reveal a problem with the theory itself. So doesn't a wave being infinite fit that problem?

[pioneer]

We can simulate the uncertainty principle with photography. If we took a photo of an action scene, with the shutter speed of the camera too slow, we will get motion blur. Motion blur is that fuzziness around moving objects in pictures What this motion blur does is create uncertainty in position, since the object appears to exist within a range of places, instead of one sharp position in distance. In the photo below, the water has position uncertainty.

 

 

But on the other hand, if we chose the correct shutter speed and stop the action, so we know exact position, but we can't tell momentum. The boys are in motion and we know where they are in position but we can't tell how fast or how much momentum they have.

 

Relative to the uncertainty principle, the phenomena is loosely related to this photography effect. The operative variable is time (shutter speed). If time is too slow in the camera, there is extra time in the photo. This is seen as uncertainty in distance or blur, due to the integration of space-time and position expressing the extra time that camera does not pick up. When there is no extra time and the shutter speed matches the motion speed, we get only sharp distance (without time), but the lack of time makes it hard to know the momentum because momentum needs time as a variable in the equation.

 

There is a work around the uncertainty principle using time. If instead of a still camera, we used a movie camera, with each frame at the correct shutter speed, to stop the boys, each frame will tell exact position. While the composite of the frames or the movie would tell us the momentum in the action. So, we look at the momentum in the movie, and pick a frame for exact position. What the movie has done is add time to the still frame.

Edited July 6, 2009 by pioneer

[Sisyphus]

That's not really an illustration of the UP, though. It might be a rough analogy, but it would be easy to take it too far and get the wrong idea. A blur doesn't show an uncertainty or indistinctness of position, it shows that the object was in all of those positions (which may as well be precisely determined) during the time in which the shutter was open. It's just like overlaying a series of precise position measurements. On the quantum scale, the water really would occupy a range of position, and the boys would not have a momentum. It would not just be a limitation of the equipment.

[swansont]

That's not really an illustration of the UP, though. It might be a rough analogy, but it would be easy to take it too far and get the wrong idea. A blur doesn't show an uncertainty or indistinctness of position, it shows that the object was in all of those positions (which may as well be precisely determined) during the time in which the shutter was open. It's just like overlaying a series of precise position measurements. On the quantum scale, the water really would occupy a range of position, and the boys would not have a momentum. It would not just be a limitation of the equipment.

 

I agree. The HUP basically says that no matter how good your camera, the boys will always be indistinct. if you zoom in, they are localized (you know they are in the frame), and they will be blurry because they can't have a well-defined momentum.

[Severian]

So each ocean wave above is 3-dimensional too, but it only stretches "infinte" along one dimension: horizontally. And its length is just as limited on the frontward-rearward axis.

 

I think you have a bit of a misunderstanding here. Waves in the ocean are 2-dimensional in the sense that the height of the wave depends only on 2 coordinates. For particle states, the magnitude of the wavefunction depends on 3-coordinates (4 if you count time), just as the temperature in a room depends on three coordinates.

[Baby Astronaut]

I think you have a bit of a misunderstanding here. Waves in the ocean are 2-dimensional in the sense that the height of the wave depends only on 2 coordinates. For particle states, the magnitude of the wavefunction depends on 3-coordinates (4 if you count time), just as the temperature in a room depends on three coordinates.

A wave of 2 coordinates is easy to draw, I suppose. How about a wave of 3 coordinates -- how might I draw that? Google has 3d wave images, but I'm not sure what I'm looking at.

 

 

(Heyy..I still got the bottom half of post #44 to get answers for Thanks )

[swaha]

did u mean that we are uncertain not by laking instruments but shall always be? but why cant we measure correctly if we can make an errorfree instrument? please explain.