[Soul Surfer]

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.

[yor_on]

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.

[Pmb]

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. 

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

[yor_on]

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.

[Bill S (OP)]

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.

[Soul Surfer]

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

[Pmb]

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

[yor_on]

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?

[yor_on]

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]

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]

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

[Soul Surfer]

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]

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]

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

A plane wave: has .  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]

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 (OP)]

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]

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]

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]

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]

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?