The Naked Scientists

The Naked Scientists Forum

Author Topic: How does momentum space differ from ordinary space?  (Read 16984 times)

Offline Bill S

  • Neilep Level Member
  • ******
  • Posts: 1802
  • Thanked: 11 times
    • View Profile
Please can someone help me to get my head around the concept of momentum space?
Simplest possible language, please.  ???


 

Offline JP

  • Neilep Level Member
  • ******
  • Posts: 3366
  • Thanked: 2 times
    • View Profile
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).
 

Offline Soul Surfer

  • Neilep Level Member
  • ******
  • Posts: 3345
  • keep banging the rocks together
    • View Profile
    • ian kimber's web workspace
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.
 

Offline Soul Surfer

  • Neilep Level Member
  • ******
  • Posts: 3345
  • keep banging the rocks together
    • View Profile
    • ian kimber's web workspace
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 »
 

Offline Soul Surfer

  • Neilep Level Member
  • ******
  • Posts: 3345
  • keep banging the rocks together
    • View Profile
    • ian kimber's web workspace
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.
 

Offline Bill S

  • Neilep Level Member
  • ******
  • Posts: 1802
  • Thanked: 11 times
    • View Profile
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.  :)
 

Offline MikeS

  • Neilep Level Member
  • ******
  • Posts: 1044
  • The Devils Advocate
    • View Profile
How does momentum space differ from ordinary space?
« Reply #6 on: 11/12/2011 07:09:02 »
Soul Surfer

Thanks for the explanation.
 

Offline Bill S

  • Neilep Level Member
  • ******
  • Posts: 1802
  • Thanked: 11 times
    • View Profile
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.   
 

Offline JP

  • Neilep Level Member
  • ******
  • Posts: 3366
  • Thanked: 2 times
    • View Profile
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.
 

Offline Bill S

  • Neilep Level Member
  • ******
  • Posts: 1802
  • Thanked: 11 times
    • View Profile
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?
 

Offline JP

  • Neilep Level Member
  • ******
  • Posts: 3366
  • Thanked: 2 times
    • View Profile
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. 
 

Offline yor_on

  • Naked Science Forum GOD!
  • *******
  • Posts: 11978
  • Thanked: 4 times
  • (Ah, yes:) *a table is always good to hide under*
    • View Profile
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?
 

Offline JP

  • Neilep Level Member
  • ******
  • Posts: 3366
  • Thanked: 2 times
    • View Profile
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. 
 

Offline yor_on

  • Naked Science Forum GOD!
  • *******
  • Posts: 11978
  • Thanked: 4 times
  • (Ah, yes:) *a table is always good to hide under*
    • View Profile
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?
 

Offline Bill S

  • Neilep Level Member
  • ******
  • Posts: 1802
  • Thanked: 11 times
    • View Profile
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. 

 

Offline JP

  • Neilep Level Member
  • ******
  • Posts: 3366
  • Thanked: 2 times
    • View Profile
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.
 

Offline imatfaal

  • Neilep Level Member
  • ******
  • Posts: 2787
  • rouge moderator
    • View Profile
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 d61d84c3b77a5061c92b320a2ff706c8.gif
If 21056fa157982e3ef20648948a53eafe.gif 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. 
 

Offline JP

  • Neilep Level Member
  • ******
  • Posts: 3366
  • Thanked: 2 times
    • View Profile
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 d61d84c3b77a5061c92b320a2ff706c8.gif
If 21056fa157982e3ef20648948a53eafe.gif 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 e0c504e6c1135b550b0f832a4124a59b.gif.  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.
 

Offline Pmb

  • Neilep Level Member
  • ******
  • Posts: 1838
  • Physicist
    • View Profile
    • New England Science Constortium
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

 

Offline Bill S

  • Neilep Level Member
  • ******
  • Posts: 1802
  • Thanked: 11 times
    • View Profile
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.  :)
 

Offline Soul Surfer

  • Neilep Level Member
  • ******
  • Posts: 3345
  • keep banging the rocks together
    • View Profile
    • ian kimber's web workspace
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.
 

Offline yor_on

  • Naked Science Forum GOD!
  • *******
  • Posts: 11978
  • Thanked: 4 times
  • (Ah, yes:) *a table is always good to hide under*
    • View Profile
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.
 

Offline Pmb

  • Neilep Level Member
  • ******
  • Posts: 1838
  • Physicist
    • View Profile
    • New England Science Constortium
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
 

Offline yor_on

  • Naked Science Forum GOD!
  • *******
  • Posts: 11978
  • Thanked: 4 times
  • (Ah, yes:) *a table is always good to hide under*
    • View Profile
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 »
 

Offline Bill S

  • Neilep Level Member
  • ******
  • Posts: 1802
  • Thanked: 11 times
    • View Profile
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.
 

The Naked Scientists Forum

Re: How does momentum space differ from ordinary space?
« Reply #24 on: 20/12/2011 22:53:25 »

 

SMF 2.0.10 | SMF © 2015, Simple Machines
SMFAds for Free Forums
 
Login
Login with username, password and session length