How does momentum space differ from ordinary space?

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Offline Bill S

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Please can someone help me to get my head around the concept of momentum space?
Simplest possible language, please.  [???]

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Offline JP

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How does momentum space differ from ordinary space?
« Reply #1 on: 10/12/2011 04:26:33 »
Bill, are you comfortable with position space?  That's when you describe something in terms of how much of it is at each position in space. 

Momentum space is the same thing, except you describe how much there is with each possible momentum.

It's useful because often times you can analyze things more easily in momentum space than in position space (particularly when dealing with waves).

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Offline Soul Surfer

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How does momentum space differ from ordinary space?
« Reply #2 on: 10/12/2011 09:37:44 »
Thank you for asking this question. In checking out my understanding with other references before I replied I have learned something important that I have in fact been seeking for a long time.

First let me agree with JP above that is a very good simple start but I would like first like to extend it a little bit.  we live in a dynamic universe and absolutely everything is moving there is essentially no such thing as a stationary object.  in normal space we define something by where it is (relative to something else of course) AND when it was there.  Because it will have moved on later.  In momentum space we  describe something by how fast and in what direction it is moving with respect to something else it is to a certain extent the other side of the dynamics of the universe.

This can be very useful in high energy physics experiments (like the LHC)  which happen in very tiny volumes and over incredibly short time periods so you can only measure what went in in what direction and with what energy and what came out with what direction and what energy. a bit like a snooker or pool table break with a visual barrier over the collision area so you cant see that bit.

But that's just the start there are more important thing than that. and I will continue in a new reply.
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Offline Soul Surfer

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How does momentum space differ from ordinary space?
« Reply #3 on: 10/12/2011 10:27:39 »
In the nature of Quantum physics and the wave particle duality there is a very fundamental relationship between things measured in noormal space/time and momentum space.

Firstly let me introduce and explain something about waves that most people appreciate but do not know by using a familiar analogy, sound waves and music.  A particular musical note in its simplest form is just a wave at a particular frequency this can be described as a periodic pattern in air pressure or movement that starts and stops at a particular time. As the note moves away from the instrument you can visualise it as a packet of waves. I could in theory describe music in terms of all the waves as they move away from the source.  That is precisely what a CD recording does and we can listen to music, but it is a very complicated process and requires a lot of data.  Now think of a piano or organ I have a number of keys and each of them generates a particular frequency for as long as I press the key.  I can think of these keys as representing music in a different way (a transform process) and this is exactly the way music is written down with each of the notes showing what frequency the note is and how long it lasts.

There is a mathematical process that can take the CD version and convert it into something like the written music with all the frequencies and their durations noted rather than describe the waves in detail.  It is called a Fourier transform.  it can be used to make the music much simpler to describe digitally and is essentially what an MP3 is. That is the way all the music fits into Ipods and allows you to get hundreds of tunes onto a CD.

Now come back to the wave particle duality.  They are two different ways of expressing the same thing.  The particle is in some ways a shorthand version of describing the full wave.  Quantum theory states that ALL particles and composite lumps of things even you! have a wave description.  This wave description has a wavelength that is measured in terms of its momentum, that is the product of its mass times its velocity plus a bit related to its contained energy.  The more momentum it has the shorter the wavelength gets. most large objects, even if they are perfectly still as far as you are concerned have so much momentum that their wavelengths are absolutely tiny.  It is only when you get to very tiny light objects like electrons and photons which have no mass but do have contained energy that this wavelength can be important.

Now there is a relationship between the normal space (particle) expression of things and the momentum space (wave ) description of things and this is exactly the same as music the Fourier transform.  Now this should not be a surprise because the Fourier transform is one of the most fundamental things about any wave motion.  This is a very deep relationship that most people including many scientists do not fully appreciate or exploit in their understanding of physics.
« Last Edit: 10/12/2011 18:37:37 by Soul Surfer »
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How does momentum space differ from ordinary space?
« Reply #4 on: 10/12/2011 10:52:05 »
As a final note I need  to ad a couple of important and interesting facts that are not obvious. 

I can hear people asking why don't we have to include real positions in our momentum space description because all the waves are moving in our real space.  The answer is that in the full wave description all the waves go on to infinity in space and time.  The real position is in fact built into the wave function.  OK there is most of the universe where the particle isn't and it doesn't play any part in what happens but it does help us to understand entanglement because in the wave expression every individual particle in the entire universe exists in the entire universe.

One other simple fact is that if ever a particle with mass had totally zero momentum and was stationary it would not be possible to tell where it was and in fact it is equally probable that it is anywhere in the entire universe.  Physicists can create Bose Einstein condensations of similar atoms and molecules at extremely low temperature that are of a finite and observable size where these atoms exist  together and jointly within this space and it is not even theoretically possible to distinguish between them or where they are individually but it is possible to detect and identify the size and shape of the condensation.
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Offline Bill S

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How does momentum space differ from ordinary space?
« Reply #5 on: 10/12/2011 13:00:01 »
Thanks for that fantastic explanation!  After just one reading I can kid myself that I undrstand it.  Print it and read it a few times, and who knows, I may get my head round it.  [:)]

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Offline MikeS

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How does momentum space differ from ordinary space?
« Reply #6 on: 11/12/2011 07:09:02 »
Soul Surfer

Thanks for the explanation.

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Offline Bill S

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How does momentum space differ from ordinary space?
« Reply #7 on: 15/12/2011 17:26:11 »
I needed to clear up the "momentem space" thing, at least a bit, in case it had any significant bearing on the train of thought I was trying to follow.  It went something like this:

The uncertainty principle says that we cannot know precisely the position and momentum of a particle at the same time; but can a particle actually be said to have a precise position, momentum or energy? 

If a particle can, for example, be said to have a precise position, then, in principle, it would be possible to know the precise position of every particle in the Universe, and there would be no positional uncertainty.  Would that not dispense with virtual particles?

Similarly, if a particle’s energy had a precise value, it would, in principle, be possible to know the precise energy content of the Universe, and there would be no zero-point vacuum energy.

Uncertainty must, therefore, be intrinsic to matter and energy, and when authors talk of measuring (e.g.) position precisely, this must be an approximation.   

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Offline JP

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How does momentum space differ from ordinary space?
« Reply #8 on: 15/12/2011 21:38:52 »
Well, there are a few things to consider when talking about the uncertainty principle.  The first is that, like all phenomena, it takes place within a model, and that model isn't a theory of everything.  In the case of the uncertainty principle, it works within all our models of quantum mechanics, and so far as we know (based on tests and theory), those models are good only up to a certain resolution, based on the Planck scale.  Most famously, below the Planck length, the theory is going to need modification.

However, if you work within the limits of the model, particles behave like points, meaning that the theory places no limits on how small they can be (and how finely resolved the energy or momentum can be).  This of course doesn't apply beyond the Planck scale, since the entire model doesn't apply beyond that scale. 

Finally, there's the experimental side of things.  In practice, we can't measure position infinitely precisely, or even down to the Planck scale, due to physically practical detectors.  The same goes for precision in measuring energy or momentum.  So in practice, we haven't come to the limits of the theory yet, let alone gone beyond to see if infinite precision is really allowed.

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Offline Bill S

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Re: How does momentum space differ from ordinary space?
« Reply #9 on: 18/12/2011 01:54:27 »
I'm not sure I know exactly what you are saying here, JP.  Could it be that we don't actually know if a particle has (e.g.) a precise position?

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Offline JP

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Re: How does momentum space differ from ordinary space?
« Reply #10 on: 18/12/2011 14:40:43 »
What do you mean by "know?"  My post above had three answers:

1) In terms of the theory of our best models so far, there is no limit on how precisely we can know position of a fundamental particle.  (Unless we know momentum, since the HUP always holds.)  In other words, the theory has allowed solutions for a fundamental particle that are size zero. 

2) In terms of experiment, there is a limit to what we can actually measure and test.  No matter how small we go, there doesn't seem to be a lower limit, but we're limited by what we can practically build. 

3) In terms of where our model is expected to need a tune-up, we expect that we could talk about precise measurements in terms of the current theory down to about the Planck length, which is 10-35 m, but we're nowhere near that in terms of experimental measurements to test the theory yet. 

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Offline yor_on

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Re: How does momentum space differ from ordinary space?
« Reply #11 on: 18/12/2011 15:11:22 »
I wonder how precise we can get a atomic clock, or some astronomical equivalent, to become? I keep looking at NIST and I can't help but getting impressed by their accuracy so far, but we still seem to have quite some bit to go.

What do you think JP, do you think we ever will come close to Planck scale? And how, if you would make a educated guess, would you expect us to get there?
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Offline JP

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Re: How does momentum space differ from ordinary space?
« Reply #12 on: 18/12/2011 18:05:40 »
Its difficult to predict where we'll be in 100 years in terms of experiments.  Based on current technology, we're a long way off from getting to the Planck scale in energy, time or length (the Planck mass is very easy to obtain, however.) 

I suspect what will happen first is that we'll develop theories that describe sub-Planck dynamics and figure out indirect ways to test them by experiment or to check them by observation of the universe rather than needing to actually measure sub-Planck lengths, for example. 

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Re: How does momentum space differ from ordinary space?
« Reply #13 on: 18/12/2011 20:16:28 »
Yeah, it's probably the way it will be done. The thing that bothers me with this kind of indirect evidence is the way they increasingly seems to rely on the mathematical probability of it existing. On the other hand, maybe there is no better way to define it as we come closer to those scales? The really esoteric stuff for me comes when one attach sigma levels to it, as that must presume that one somehow 'know' a standard, to be able to measure deviations from it.  Doing it this way it becomes less and less transparent to a layman, or, as a guess also physicists, mathematicians as I would expect arguments around what that 'standard' should be seen as, as the experiments becomes more and more complicated/convoluted.

Or is there another way to mathematically define sigma levels?
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Offline Bill S

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Re: How does momentum space differ from ordinary space?
« Reply #14 on: 18/12/2011 21:20:36 »
Sorry if I'm being a bit naïve, here; I blame it on age/ignorance, and who's going to argue with that?

No problems with your points 2 & 3, but, just to clarify that point 1 says that in theory we could measure an absolute position for a particle.  If this is so, should we not, in principle, be able to identify an absolute position for every particle in the Universe? 

In no way am I suggesting we could do this practically. 


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Offline JP

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Re: How does momentum space differ from ordinary space?
« Reply #15 on: 18/12/2011 22:16:12 »
Yes, you could one-by-one measure each particle in the universe to establish an absolute position for each one.

No, you couldn't know all those positions simultaneously because you'd have to travel between them.  Once you measured a particle and left to measure a new one, that first particle would start moving and interacting with the rest of the universe, so you wouldn't know its position any longer.

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Offline imatfaal

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Re: How does momentum space differ from ordinary space?
« Reply #16 on: 19/12/2011 11:39:57 »
JP - could you clarify/explain. 

I always thought that you could not have absolute accuracy of any of the HUP variables.   For a start it would screw up the inequality [tex]\sigma x\sigma\rho  \geq  \frac{\hbar}{2} [/tex]
If [tex]\sigma x= 0[/tex] you are never going to get that to work out - it might imply an infinite variation in momentum, but that is not really acceptable.

Secondly - the simplistic version of Feynman path integral will give the time before any particle will quantum tunnel to a different place as linearly proportional to the uncertainty in the position; thus if the uncertainty in position is zero then the probability of the particle still being there is zero. 
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Offline JP

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Re: How does momentum space differ from ordinary space?
« Reply #17 on: 19/12/2011 12:24:24 »
JP - could you clarify/explain. 

I always thought that you could not have absolute accuracy of any of the HUP variables.   For a start it would screw up the inequality [tex]\sigma x\sigma\rho  \geq  \frac{\hbar}{2} [/tex]
If [tex]\sigma x= 0[/tex] you are never going to get that to work out - it might imply an infinite variation in momentum, but that is not really acceptable.

A plane wave has [tex]\sigma_p=0[/tex].  Something with no position uncertainty is a delta function in position.  Both are legitimate solutions to the model, even if its questionable if they can be physically realized.  (The plane wave must be infinitely large in space, while the delta function is smaller than the Planck scale.)

I don't think the Feynman path integral issue is a problem.  What I believe its saying is that knowing position accurately means this has a huge spread in momentum, so that once you measure it, it's almost certainly going to move.

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Offline Pmb

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Re: How does momentum space differ from ordinary space?
« Reply #18 on: 19/12/2011 13:07:27 »
Please can someone help me to get my head around the concept of momentum space?
Simplest possible language, please.  [???]
Ordinary space has three dimensions where things are placed. Momentum space is quite different. It's an abstract set of a set of momentum. That's a poor explanation on  my part since it's been a while since I had to define such abstract spaces. Here is a good definition - http://en.wikipedia.org/wiki/Momentum_space

Pete


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Offline Bill S

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Re: How does momentum space differ from ordinary space?
« Reply #19 on: 20/12/2011 00:06:26 »
Thanks, Pete.
 
Actually, that's where I started looking!   It lost me after the first few sentences, which is why I was looking for something simpler.

From the way this thread has developed, I think I'm in over my head.  :)

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Offline Soul Surfer

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Re: How does momentum space differ from ordinary space?
« Reply #20 on: 20/12/2011 09:06:44 »
Thanks for adding that link Pete.  I should have put it with my explanation at the top of the page.  I did make use of this link in writing up my from the roots explanation.  but I needed much more because most people do not really know about Fourier transforms. 

It also helps to link in the uncertainty principle and the wave-particle duality.
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Offline yor_on

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Re: How does momentum space differ from ordinary space?
« Reply #21 on: 20/12/2011 14:35:55 »
Yeah, the math is sort of scary for us laymen :)

When I read SoulSurfer I started to wonder if you could describe the Fourier transform in images and I found  An Intuitive Explanation of Fourier Theory.

It's just as SoulSurfer says, you 'translate/transform' a wave (wave function) into a simpler representation keeping the information that describes what you're interested in, as a sound or a image. In quantum mechanics, momentum is defined as an operator on the wave function.

Here you can see how that is defined in QM as related to Heisenberg's Uncertainty Principle. And if you ever used a JPEG and wondered how they can compress a image into such a small size then you wondered about Fourier transforms Fourier analysis. But the trickiest part is to understand what a momentum mean for a wave (wave function), and how it is represented intuitively.
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Offline Pmb

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Re: How does momentum space differ from ordinary space?
« Reply #22 on: 20/12/2011 14:51:54 »
You're welcome good sir Soul Surfer. :)

One of things I love about posting in forums is that when you see someone who says I was helpful it's a wonderful feeling.

You can help me work on the site if you wish. You can tell me what is needed, where it is needed, if a page or group of pages needs to be restructed, if an explanation leaves somethning to be desired etc. I'd love the input!  :)

Do to you desire you mentioned I think my next entry will be on Fourier Transforms. It will take a lot of time to create it since it will be very thorough. I.e. I have some reviewing to do since it's been a long time since I was fluent in the subject. In fact I have an entire book on the subject, i.e. Fourier Series, by Georgi P. Tolstov, Dover Pub.  Don't be fooled by the name of the book since it covers everything about the subject, i.e. it contains the subjects Fourier Series, Fourier Integrals and Fourier Transforms. Fourier Integrals are used in optics so I plan on (re) learning optics, and, as most of us know, it's used a lot in quantum mechanics, which I'm fairely fluent in Whew! That's going to be a lot of work. That's why I'd love the input. Are you up to it? You don't need to help at anytime excecpt when you have some input you'd like to inject, perhaps a page you yourself wrote, or when you're bored or just filled with energy that you're raring to go.  :D

Dear yor_on,
Thanks for that input. The level of the math was neccessary because I created pages to explain things as they arose in forums (I can't very well post integrals in a news forum) and the answers required a high level of mathematics and physics (graduate level stuff) to answer properly. I'd love as much input as you wish to add. I.e. how can I help make these pages accessible to the layman?

Best wishes

Pete

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Re: How does momentum space differ from ordinary space?
« Reply #23 on: 20/12/2011 15:34:14 »
Hmm, Pete, to me it seems as it is the wave particle duality, again :)

So if you want to explain it simple describing what a momentum space is for a wave (function) then you better start with how we defined that a wave could be seen as having a momentum. A thing that both particles and waves seems to share. So some De Broglie to start with and just enough math to see what the he* it means. That should mean that you will need to explain Planck's constant/scales too, and how he reached them :) and how the combination of waves and momentum came to be.

0uch, I know, it's a lot but it's a good start. And Plank is one of my favorites in physics, he's still very impressive to me. From there you better be very careful if you want us to see what  the he* the mathematical definition of a momentum space is. I think it's going to be tricky, at least if you describe it with time as a necessary component, which it should be. It seems quite complicated to explain.
« Last Edit: 20/12/2011 15:39:13 by yor_on »
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Offline Bill S

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Re: How does momentum space differ from ordinary space?
« Reply #24 on: 20/12/2011 22:53:25 »
Quote from: SS
One other simple fact is that if ever a particle with mass had totally zero momentum and was stationary it would not be possible to tell where it was and in fact it is equally probable that it is anywhere in the entire universe.

Why is this?  Intuitively it would seem that a stationary particle should be the easiest to locate.

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Offline Soul Surfer

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Re: How does momentum space differ from ordinary space?
« Reply #25 on: 20/12/2011 23:55:31 »
I was dragged away before I could complete the bit on the wave particle duality and the uncertainty principle.

The really important think is when you describe something as a wave, waves always go off to infinity both mathematically and physically.  OK they mostly become vanishingly small at large distances but they may not always do so, so in reality all "particles" in their wave description exist in all the universe.

Now there is also one thing that people forget, there IS something that can exceed the speed of light and that is the phase velocity of a wave, this can travel infinitely fast given the correct circumstances.  For example microwave radio signals in cut off waveguides and quantum particles tunnelling through barriers.  The important fact that prevents any of the rules being broken is that it is not possible to transfer any information faster than the speed of light.

However if you have two entangled particles you can visualise them as two linked entities that exist at different positions and momentum and the measurement of the entangled property defines the phase of the wave function linking them this instantly forces the phase of the other particle to be defined but no information can be transferred because you could not possibly know what it was before you defined it.

The uncertainty principle just defines how well you can describe the most probable position of the wave peak which is called the particle.

If you think of absolutely everything as being essentially waves that are in some cases confined to locations as "particles" and all these waves have been inextricably linked ever since the big bang,  quantum mechanics becomes quite rational and normal.

The problem is that we normally look at the world as positions and times and not as momentum waves in momentum space
« Last Edit: 21/12/2011 00:00:01 by Soul Surfer »
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Re: How does momentum space differ from ordinary space?
« Reply #26 on: 21/12/2011 12:33:43 »
So if you want to explain it simple describing what a momentum space is for a wave (function) then you better start with how we defined that a wave could be seen as having a momentum.
It's the particle which carries the momentum, not the wave. There is one thing and one thing only to remember about the wave aspect of quantum mechanics and that's that the meaning of the wave is that it tells you is that the square of the amplitude of the wave function indicates the probability of where the particle will be found. The actual value has to do with integrating the wave function over the region you're interested in.

Bill - A stationary particle has a well defined position which makes the momentum have an ill-defined position. However, in reality, it can't be just anywhere in the universe. Quantum mechanics as you know it is only an approximation. Relativistic quantum mechanics is more accurate. Since nothing can travel faster than light, the particle can only travel so far before you attempt to measure its location again, and the position you find it at will be at a finite position near to you .... unless you wait a trillion years for you next measurement. :)

Best wishes

Pete

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Offline yor_on

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Re: How does momentum space differ from ordinary space?
« Reply #27 on: 21/12/2011 13:29:30 »
Bill, to know a 'global momentum' one would first need a global definition of a 'speed/velocity'. According to the experiments, and this being my view, that's not possible to know. You can define a particle 'at rest' relative something, like the accelerator, the particle being constrained, 'at rest' in a magnetic field, but is that particle 'at rest' with the universe?

Maybe it is :) I'm not sure at all here what to think, but if it is we find two definitions now, as something being 'at rest' relative something else, as compared to something being 'still' and so 'at rest' universally/globally. While our first particle being 'at rest' it can still have a different 'uniform speed' relative some other uniformly moving thing, as another planet maybe? Having and using its own LHC accelerator to measure a particle they too constrain in some magnetic bottle, being 'at rest' with the accelerator before accelerating. Maybe my analogy is flawed, but consider what being 'at rest' means, to see the point. If I can't define a 'global null speed' then the only thing left is using being 'at rest'. But that also brings with it a uncertainty relative all 'speed' except 'c' as both 'LHC:s' here defined their particle as being 'still', although they clearly were not 'at rest' relative each other.

Described as a wave function you can't have all 'properties' of that particle existent at the same time. So the closer you define its position the more indefinable its momentum becomes according to HUP, with that indefinable momentum spreading it out all over space as if it was a wave. It's also about its mathematical definition, where we are used to think of 'particles' as the thingies creating touchable objects, as molecules and matter, mathematically it becomes something else.

"In 1926, just weeks after several other physicists had published equations describing quantum physics in terms of matrices, Erwin Schrödinger created quantum equations based on wave mathematics, a mathematical system that corresponds to the world we know much more then the matrices. After the initial shock, first Schrödinger himself then others proved that the equations were mathematically equivalent. Bohr then invited Schrödinger to Copenhagen where they found that Schrödinger's waves were in fact nothing like real waves. For one thing, each particle that was being described as a wave required three dimensions. Even worse, from Schrödinger's point of view, particles still jumped from one quantum state to another; even expressed in terms of waves space was still not continuous. Upon discovering this, Schrödinger remarked to Bohr that "Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business."

Unfortunately, even today people try to imagine the atomic world as being a bunch of classical waves. As Schrödinger found out, this could not be further from the truth. The atomic world is nothing like our world, no matter how much we try to pretend it is. In many ways, the success of Schrödinger's equations has prevented people from thinking more deeply about the true nature of the atomic world."  What is Quantum Physics?
« Last Edit: 21/12/2011 13:46:05 by yor_on »
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Re: How does momentum space differ from ordinary space?
« Reply #28 on: 21/12/2011 14:01:31 »
Maybe Pete? You need to explain that one in more detail for me to get how you mean, but a electron can behave as a wave.. Also you have light defined as having a momentum?
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Re: How does momentum space differ from ordinary space?
« Reply #29 on: 21/12/2011 14:08:16 »
Bill, you can draw analogies here to water waves, since these particles are getting all their interesting properties from their waviness.  One important difference to keep in mind is that in a water wave, the water is physically waving.  In a quantum wave, the height of the wave basically tells you the probability of finding the particle if you look there.  You can't measure the whole wave at once.

So you can ask what kind of water wave represents a particle whose position you know exactly.  Well, if drop a rock into the water, you know (pretty much) exactly where it hit the water.  The wave coming out is a circular wave, traveling in all directions.  If I ask you "what direction is the wave going?" you won't be able to pick a single direction, or even a range of directions, as its spreading in all directions equally.  In QM, if you tried to measure the way the particle was moving away from that initial point, you'd be equally likely to see it moving in any direction.  The wave, by the way looks like this:


Its much harder to intuitively think of a wave that only has one direction (momentum).  You can think of a wave coming towards a flat beach.  The wave is infinitely wide in the direction parallel to the beach.  Its peaks and troughs are also equally spaced.  It will always maintain this equal spacing and move towards the peach without changing its overall shape or spreading out. 
It looks something like this:

You can imagine it moving from left to right so it definitely has a direction.

What's not always so clear is that if block part of the plane wave, it starts not quite moving perfectly left to right.  The wave near the edges of the obstruction starts spreading out around the obstruction, a process called diffraction.  I won't draw this one.  I'll go to Wikipedia for help.  Check the diffraction figures: http://en.wikipedia.org/wiki/Ripple_tank#Diffraction.  The same happens if the wave isn't infinitely wide: near its edges, it starts spreading out.  So the wave overall doesn't have one single direction anymore, although it might have a very strong direction.  As you make the wave narrower and narrower, the spreading becomes more and more important to its overall direction.  If you narrow it down to a tiny point, the wave coming out of that point looks like the circular wave I showed above. 

I'm going to borrow someone else's explanation again, because its pretty cool.  Light (in fact, all waves) have similar behavior.  You can make a laser beam pretty much a plane wave (of course, it's not infinitely wide, but you can make it spread out very little over the meter or so it travels in the lab).  If you shine it on a screen that has a small hole cut into it, it will start to spread out after that hole.  That's because the hole described the position of that wave very well, so it will take a wider range of directions.  As you make the hole smaller, it spreads out more, since you know the position better.  There's a nice java applet where you can play with this, located here: http://micro.magnet.fsu.edu/primer/java/diffraction/basicdiffraction/

You can make the transition to quantum waves by thinking along similar lines.  For a particle's wave to be traveling in one particular direction, it has to be very wide in space.  If you force the wave to travel through a tiny hole (which means you know the particle traveled through that tiny hole so you knew its position well), the wave coming out of the hole spreads into many directions, so you don't know its momentum well.  At the end of the day, the wave's spread in position tells you where the particle is likely to be if you try to measure it's position, so the plane wave means the particle could be anywhere in space.  The spread of a particle in direction tells what momentum you'll measure if your detector can measure momentum.  A circular wave has a highly undetermined momentum. 

I swept one thing under the rug here, which isn't really a big deal for this hand-waving explanation.  When I talked about momentum, I wasn't changing the magnitude of momentum, which is how fast the wave peaks are moving.  I was only changing the direction.  Both can vary when talking about QM.  But again it doesn't make a big difference to understand the basics.

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Offline imatfaal

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Re: How does momentum space differ from ordinary space?
« Reply #30 on: 21/12/2011 17:13:22 »
Quote from: SS
One other simple fact is that if ever a particle with mass had totally zero momentum and was stationary it would not be possible to tell where it was and in fact it is equally probable that it is anywhere in the entire universe.

Why is this?  Intuitively it would seem that a stationary particle should be the easiest to locate.

But QM is not intuitive - a particle with a momentum defined absolutely will either be anywhere in the universe or fill the the universe. 

I think that rules out absolute precision - but JP (who does know a lot more than me) says that there are circumstances in which  [tex]\sigma \rho[/tex] can be zero and I am still trying to understand them (or at least argue with them)
« Last Edit: 21/12/2011 17:16:48 by imatfaal »
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Offline Soul Surfer

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Re: How does momentum space differ from ordinary space?
« Reply #31 on: 21/12/2011 22:49:52 »
The problem is that you are trying to think in terms of particles whereas it is really the waves that are the most fundamental thing in our universe which consists only of energy and momentum expressed as waves.  These are the only things that are truly conserved.  Particles, time and space are all products of the existence and interaction of energy and momentum.

That does not mean that particles, time and space are not useful concepts for understanding reality.  It does mean that insisting that they are the main or only thing will throw up inconsistencies.

Experiments with very low temperatures and Bose Einstein condensations of atoms shows them clearly as macroscopic objetcs many orders of magnitude bigger that the atoms of which we are familiar.  Remember we are not just dealing with individual particles here but whole composite atoms containing protons neutrons and electrons with significant energy levels but the collective item has a very long wavelength wave function because of their extremely low momenta of the boson atoms (ie unity or zero total spin) with respect to each other.

I said elsewhere that fundamental physicists should learn more chemistry in the hope that they would understand better how quantum symmetries really break.  I think all physicists studying quantum theory should learn the basics of communications and information theory including the concept of time-bandwidth product because this shows clearly the relationship between particles (information bits) and waves (amplitudes and phases) and shows that the uncertainty principle is simply a statement of this fundamental information relationship and not something that someone plucked out if the air to explain experimental errors.
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Offline JP

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Re: How does momentum space differ from ordinary space?
« Reply #32 on: 22/12/2011 00:44:22 »
I said elsewhere that fundamental physicists should learn more chemistry in the hope that they would understand better how quantum symmetries really break.  I think all physicists studying quantum theory should learn the basics of communications and information theory including the concept of time-bandwidth product because this shows clearly the relationship between particles (information bits) and waves (amplitudes and phases) and shows that the uncertainty principle is simply a statement of this fundamental information relationship and not something that someone plucked out if the air to explain experimental errors.

Isn't the fundamental relationship just between Fourier conjugate variables?  Information theory is a derived science from more fundamental principles.  The math of Fourier conjugate variables is the underlying relationship that ties all the uncertainty relationships together all the way from information theory/signal processing to optics to acoustics to quantum mechanics.

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Re: How does momentum space differ from ordinary space?
« Reply #33 on: 22/12/2011 00:57:27 »
Quote from: SS
One other simple fact is that if ever a particle with mass had totally zero momentum and was stationary it would not be possible to tell where it was and in fact it is equally probable that it is anywhere in the entire universe.

Why is this?  Intuitively it would seem that a stationary particle should be the easiest to locate.

But QM is not intuitive - a particle with a momentum defined absolutely will either be anywhere in the universe or fill the the universe. 

I think that rules out absolute precision - but JP (who does know a lot more than me) says that there are circumstances in which  [tex]\sigma \rho[/tex] can be zero and I am still trying to understand them (or at least argue with them)


A plane wave: [tex]\exp(\mathrm{i}\mathbf{p}_0\cdot\mathbf{x})[/tex] has [tex]\sigma_0=0[/tex].  You can argue that its not a physically realizable solution (to be properly defined it's over infinite space for a precise energy).  But a plane wave is an extremely good approximation to some things. 

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Re: How does momentum space differ from ordinary space?
« Reply #34 on: 22/12/2011 13:04:46 »
Yes I agree that just comes from the mathematics JP but dealing with the same mathematics in a classical and observable situation makes it much easier to accept the same mathematics in the quantum domain,
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Offline Bill S

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Re: How does momentum space differ from ordinary space?
« Reply #35 on: 23/12/2011 13:40:04 »
Thanks again, folks.  There's quite a lot to think about here, but there are also some really good pointers to help get the thinking in the right direction.

Thinking may have to give way to some good single malt over the next few days, with any luck.

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Offline Pmb

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Re: How does momentum space differ from ordinary space?
« Reply #36 on: 23/12/2011 14:19:58 »
Maybe Pete? You need to explain that one in more detail for me to get how you mean, but a electron can behave as a wave.. Also you have light defined as having a momentum?
You say that an electron behaves as a wave when actually it's nehaviour is guided by the wave's probability. And yes. Light does have momentum. It's an example of radiation pressure.

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Offline yor_on

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Re: How does momentum space differ from ordinary space?
« Reply #37 on: 24/12/2011 08:32:03 »
Hmm, what you are discussing is the mathematics defining a wave (like) property, right PMB, defining it from probability? Lot's of people want to define it from waves, sometimes begetting particle (like) properties. You seem to take the other approach defining wave (like) properties to particles. I've seen both and I found the particle approach more used earlier than it is now. I guess science has its fads too :)

I used to, and maybe still does, look at it from that approach. The problem I found doing so is that people gets very argumentative, wanting their outlook to be the one defining physics, including me arguing :) So I try to avoid to nail it down those days. After all, the particle wave duality do exist, very persistently, in the experiments as I understands it, the rest becoming our interpretations.
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Re: How does momentum space differ from ordinary space?
« Reply #38 on: 25/12/2011 14:41:35 »
If anybody wishes to read more on Max Planck then see http://en.wikipedia.org/wiki/Max_Planck. I'm not 100% sure of the accuracy of that page but it's a place to start.

Merry Christmass,

Pete

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Re: How does momentum space differ from ordinary space?
« Reply #39 on: 26/12/2011 15:57:10 »
Maybe Pete? You need to explain that one in more detail for me to get how you mean, but a electron can behave as a wave.. Also you have light defined as having a momentum?
Do you know the difference between a canonically conjugate pair/A set of canonical coordinates and its set of caanonical Momomentum? Since my pages are relatively advanced where do you propose to put the intro material? Before the beginning of the advanced stuff, athe the bottom where the advanced stuff is finished?

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Offline yor_on

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Re: How does momentum space differ from ordinary space?
« Reply #40 on: 28/12/2011 12:57:45 »
Woof Pete, the biggest problem with using words with precise definitions is understanding what they really mean, as seen in that specific context. Maybe we should create a mathematical section on TNS first? As a mere layman I often find the formulas assuming all too much knowledge of what the variables, functions, etc, really mean. They may have been ever so clear to the guy who wrote it, but to us seeing it,  a lot of information missing. As for canonical properties this one sums it up quite nicely I think.

"physics can be described as studying how quantities evolve with time, in particular, we are interested in studying those quantities whose measurements can be arrived at by "freezing" time at a particular moment. So the position of the rock that I just threw is one such variable, while its velocity and acceleration are not. Assuming time differentiability, knowledge of the measurements of such quantities is enough: you can differentiate them in time to give the values of other dynamical variables.

To put it more mathematically, physics is the study of a bunch of time dependent functions with are a priori independent from each other.

The goal of physics is then to find rules that associate different such functions: Newton's law of universal gravity assigns a rule of interaction via gravitational attraction between to bodies, and it only suffices to know the positions of those bodies at any given time to find the interaction.

Now, as it turns out, the rules of physics the were formulated based on experimental evidence, in many cases, can be written as differential equations on those measured quantities with second order derivatives in time. Mathematically, this says that physics can be described as the study of a system of time dependent functions whose evolution is governed by a second order ordinary differential equation. (There are, of course, exceptions to this, but in those cases we do not have canonical conjugate variables anyway.)

Now, it is well known that a second order ordinary differential equation can be uniquely solved if we provide, as initial data, the value of the function at time 0, and the value of its first derivative at time 0. This implies the well known Newtonian philosophy that knowing the position and momentum of every particle enables one to solve for their dynamics for all eternity. Now, knowledge of the first derivative is not essential to the knowledge of how the world operates, since with a complete knowledge of the functions for all time will imply knowledge of the first derivatives. Yet, the knowledge of the functions for all time is encoded simply in the knowledge of the function and its first derivative at one particular time and the laws of physics.

Here we have the canonical conjugate variables. We take as half of the variables those time-dependent, a priori mutually independent functions that are sufficient to describe the dynamics. For the other half, we take them to be the time-derivative of those aforementioned functions.

In other words, the canonical conjugate variables are those variables one arrives at from the following procedure in the study of ordinary differential equations:" From What is a good non-physics definition of canonical conjugate variables?

Or "Words used in math and science are often borrowed from the common use. "Given from God" is one of the non-technical meanings of "canonical". Specifically, "appearing in a Biblical canon".

I usually take it to mean "basic" or "simplest", another common (non-technical) meaning of the term. This common usage carries over to meany technical fields. For example, the canonical equation of a circle in Cartesian coordinates is x^2+y^2=r^2 (or maybe even the more basic equation, x^2+y^2=1).

Mathematicians and physicists have yet other meanings: canonical decompositions in math, canonical variables in quantum physics, canonical ensembles statistical physics. Each of these concepts has very precise definitions." By D H..

So, I would suggest using as simple words as possible, if  that's impossible you need to rethink the concept and see if you can find another way to describe it. But it also depends on who you want to communicate with. There are some guys here that just luve math :) and maybe, a math section would be cool?

But, only if those writing there know their formulas in depth, and actually know how to explain them, equations may look cool but without naming what they describe they're worthless to most of us here. I'm not talking about you here Pete. It's just that I've seen examples of guys going in over their depth at TNS before, bamboozling us with equations not appropriate to what they want to prove/discuss, and not able to explain how they reached their mathematical conclusions either. 





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Offline Bill S

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Re: How does momentum space differ from ordinary space?
« Reply #41 on: 28/12/2011 14:15:38 »
Quote
There are some guys here that just luve math :) and maybe, a math section would be cool?

A math section would be cool; but the coolest type would be one that made the maths accessible to "hitch-hikers" like me.

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Re: How does momentum space differ from ordinary space?
« Reply #42 on: 28/12/2011 14:58:42 »
Back to the original topic, I have something that might help (or confuse) you, Bill.  :)

I don't want to post another site's image here, so you can click on this link to see what I'm going to describe:  http://jeb.biologists.org/content/214/22/3829/F2.large.jpg  What you want to focus on are the top row of plots and the bottom row (we'll get to the middle later).  The top row shows a sound wave vs. time, which you're probably familiar with.  The height of the wave indicates the amplitude (energy) and the horizontal axis indicates time, so you have a plot of the energy of the sound wave vs. time.  The bottom row is what's called a power spectrum. 

It turns out you can generate the top plots by adding up a bunch of plots of sine waves, where a different amount of each sine wave is added.  Since we know each sine wave has a specific frequency, we can make a plot of how much of each frequency sine wave needs to be added to get the top plots.  This is what's being shown on the bottom plot.  The height of the plot indicates the amount (amplitude) of each sine wave, and the horizontal axis indicates the frequency of the sine wave.  By doing this transformation, we've gone from position space (top plots) to frequency space (bottom plots), where frequency space basically tells us the amount of each sine wave needed to construct the original waveform.

The same thing holds for position/momentum space.  The math gets worse, since position is usually 3 dimensional, and so momentum is as well.  Now instead of just frequency, the sine waves have a frequency (1 dimension) and direction (2 dimensions).  But you can do the same trick and figure out how much of each sine wave you add to get the original waveform.  If you plot this, it will be a plot of the amount (amplitude) of each sine wave vs. frequency and direction.  For a sine wave, the frequency is the magnitude of momentum and the direction is the direction of momentum.  So momentum space is just a way of saying which sine waves make up the original wave. 

-------------------------------

I'm not sure if that's clear or not, but its a hand-wavy way of getting at the mathematics relating time to frequency and position to momentum.

The middle plots in the above link are another way of writing a function in terms of position and momentum (actually time and frequency in that plot).  The horizontal axis is time, the vertical axis is frequency, and the color indicates "how much energy" there is at that time for that frequency.  These kinds of plots are called phase space plots, and they're extremely useful.  If you're interested, I can go on at length about their properties, since these are one of my research interests.  But this is a long post so I'll leave you with this: I'm sure you're familiar with one kind of phase space plot of frequency (vertical axis) vs. time (horizontal), which is a way of transcribing a sound wave into something more usable by musicians:



[Source of the last image: http://en.wikipedia.org/wiki/Sheet_music]
« Last Edit: 28/12/2011 16:19:59 by JP »

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Offline imatfaal

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Re: How does momentum space differ from ordinary space?
« Reply #43 on: 28/12/2011 16:15:44 »
Back to the original topic, I have something that might help (or confuse) you, Bill.  :)
/snipped

Helped me!  Nice post JP !
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Offline Pmb

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Re: How does momentum space differ from ordinary space?
« Reply #44 on: 28/12/2011 20:08:43 »
"physics can be described as studying how quantities evolve with time, in particular, we are interested in studying those quantities whose measurements can be arrived at by "freezing" time at a particular moment.
Can I send you an article by Karl Popper?

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Re: How does momentum space differ from ordinary space?
« Reply #45 on: 28/12/2011 21:10:40 »
Sure :)
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Offline Bill S

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Re: How does momentum space differ from ordinary space?
« Reply #46 on: 29/12/2011 18:31:42 »
Quote from: JP
Back to the original topic, I have something that might help (or confuse) you, Bill.

I'm doing my best to make it help, but I'm easy to confuse. :)

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Offline JP

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Re: How does momentum space differ from ordinary space?
« Reply #47 on: 30/12/2011 04:13:06 »
Quote from: JP
Back to the original topic, I have something that might help (or confuse) you, Bill.

I'm doing my best to make it help, but I'm easy to confuse. :)

I always find a glass of nice single malt helps with that.  :)

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Offline Geezer

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Re: How does momentum space differ from ordinary space?
« Reply #48 on: 30/12/2011 05:25:52 »
I actually understood this bit! (The words too.)




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Offline imatfaal

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Re: How does momentum space differ from ordinary space?
« Reply #49 on: 30/12/2011 10:31:59 »
Quote from: JP
Back to the original topic, I have something that might help (or confuse) you, Bill.

I'm doing my best to make it help, but I'm easy to confuse. :)

I always find a glass of nice single malt helps with that.  :)

I find that a nice glass of single malt helps with practically anything
There’s no sense in being precise when you don’t even know what you’re talking about.  John Von Neumann

At the surface, we may appear as intellects, helpful people, friendly staff or protectors of the interwebs. Deep down inside, we're all trolls. CaptainPanic @ sf.n