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Author Topic: How does momentum space differ from ordinary space?  (Read 6827 times)

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 »

Pmb

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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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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 »

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?

JP

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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.

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  3b83196e7a9cc9c91fcbaf419d4329d0.gif 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 »

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.

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.

JP

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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  3b83196e7a9cc9c91fcbaf419d4329d0.gif can be zero and I am still trying to understand them (or at least argue with them)


A plane wave: 0db6d28cee4e304ad71ee041723db487.gif has 96c9057b66f2d904e0eb9ea03ebe1f59.gif.  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. 

Soul Surfer

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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,

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.

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.

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.

Pmb

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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

Pmb

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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?

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. 






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.

JP

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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 »

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 !

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?

yor_on

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

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. :)

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.  :)

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.)




[Source of the last image: http://en.wikipedia.org/wiki/Sheet_music]

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

 

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