Exploring Photon Trajectories in Double Slit Experiments

[Ken G]

I really don't know what you are talking about, to clarify the situation, please write down the classical formulation of wave mechanics that predicts the diffraction of "large ensembles of electrons".

First you need a classical measurable. Electron energy flux density will suffice (we could use electron number flux, but my point is that we never need to think of these things as particles at all to "understand" diffraction). Now you need a wave theory. Huygen's principle works fine. Let's simplify life and just get a theory that works for electrons of a given energy (which here means a given ratio of energy flux to mass flux). The wave equation with the v of that population of electrons will work fine, where v is found from timing experiments. Now we need a concept of frequency because it's a wave theory, and here we can leave the frequency as a free parameter that the interference experiment will determine. The wave equation describes the speed of signal propagation, Huygen's principle tells us how to handle the sources and the slits, and the frequency parameter gives us the interference we need. Every one of these is a 100% classical concept, remember that we are pretending we don't even know we have particles here. Now we put them together to calculate the energy fluxes everywhere subject to the free frequency parameter, compare to experiments, and poof, both the frequency parameter drops out, and the fact that we have what we would call a correct theory for electron diffraction, and all classical.

This will all work fine as long as the electron ensemble is prepared in a uniform way, say by using a fixed potential drop to accelerate the beam (which in our present quantum understanding would say gives "monoenergetic electrons"). If you object that the theory so far only handles one voltage, we can just repeat the whole experience for other values of the voltage, ultimately tracing out the dependence of frequency on voltage. Still purely classical, and now we have the "Green's functions" for a complete classical description of any type of electron diffraction experiment on a large ensemble of electrons, so long as we know how that ensemble was generated. All perfectly mundane classical physics, we can do the whole thing imagining we have an "electron field" and no electrons at all.

That we can always do all this depends on only one thing: the correspondence principle. I'm saying the correspondence principle could be reframed thusly: "If the experiment is done in the classical limit, it will be describably by a classical theory", or "any quantum theory gives birth to a classical theory in the limit of large occupation numbers."

The Schrodinger equation is mathematically similar to a classical wave equation, but it is physically distinct.

And you trace that to its quantum nature? What in the Schroedinger equation jumps up and says "I'm a quantum"? Nothing.

The important distinction is that the energy of a classical wave is related to its amplitude, whereas the energy of a quantum wave is related to its frequency.

Again you seem to be saying there is no correspondence principle. But there is.

You can rationalize it by using mathematical tricks like taking the limit as h-->0, but those are just tricks.

Labelling something a "trick" does not make it untrue. If a procedure spawns a classical theory, then voila, you have a classical theory. I'm sorry if you feel you got tricked in the process. The Schroedinger equation for electrons spawns a nonrelativistic classical theory simply by taking the limit of h-->0 keeping h/m a constant, all it has is a different dispersion relation from the dispersionless wave equation we are most used to seeing.

However, we use devices everyday that rely on h NOT being zero, such as photoelectric detectors, flash memory and LCD monitors.

All of which is irrelevant to anything I've said. Note I never said "there is no such thing as a quantum effect that has no classical analog." What I did say is "many quantum effects do have classical analogs that we take advantage of all the time, especially when testing quantum mechanics, yet many people seem to be unaware of this fact."

Nope, that is just not correct .. it's not a matter of having the equations, it's a matter of the significance of the terms in those equations.

Again, nothing we are talking about has anything to do with the philosophical interpretation that we commonly give to quantum phenomena. The point I'm making is quite a bit simpler than that: all quantum theories spawn classical theories in the classical limit, and such classical analogs can be very useful for understanding a wide array of what we call quantum phenomena, in stark contrast to what is often said about many kinds of quantum weirdness. Most quantum weirdness is simply the fact that it seems weird to us that a wave theory applies to quanta, but since wave theories are classical, we should not say that it's weird because there's no classical analog, we should say it's weird because there is a classical analog and we didn't expect that.

As I said above, the mathematical similarity between the classical and quantum mechanical wave equations does not equate to a physical similarity.

And that just doesn't make any sense. There is no distinction between "mathematical similarities" and "physical similarities", the mathematical is all we have to understand the physical.

What you seem to be missing in all of your classically based arguments is the fundamental importance of the uncertainty principle.

Ah, the uncertainty principle-- another 'quantum phenomenon' with a perfectly classical analog. Comes up in Fourier transforms of classical fields all the time, thanks for bringing that up.

[qutoe]That is where the "wave-like" properties of massive particles originate from (in the theoretical treatment anyway), and without it, you can never get from the equations of classical mechanics to the diffraction of electrons.[/QUOTE]Well, I just told you how you could do just exactly that. Remember, we arleady agreed that "classical" does not mean "Newton's laws."

 

[Ken G]

This really is a ridiculous statement. Almost the opposite is true, everything we know about quantum mechanics comes from theoretical deductions and "guesses" based on the barest of experimental data (sometimes none, eg de Broglie waves, Dirac's antimatter deduction, Bose statistics etc)

I don't even know what you're talking about, you're saying quantum mechanics is some kind of data-poor area of physics? That's just baloney, quantum mechanical predictions confront a spectacular amount of extremely accurate data all the time. I don't even see how this wrong point would be relevant if it were right!

Where did you get the idea that experimental results enable us to "understand the quantum". They just confirm that the (spectacularly non-classical) theory is not falsified.

I guess you see some distinction in those two sentences, but I'm afraid I do not.

 

[Ken G]

Ken, reading you I get confused. Are you arguing that if there isn't a equivalence to a macroscopic explanation for a QM experiment, then it must be wrong?

Certainly not, had I meant that I would have said that. I'm saying that any quantum mechanical theory spawns a classical theory that works on the classical limit. It's really very simple, it's just the "correspondence principle." By "classical theory" I mean one that takes advantage of purely classical constructs, and could have been developed by physicists who were never even aware they were dealing with quanta. It's kind of an accident of history when exactly that happened, and when it did not.

What I am not saying is that there isn't any such thing as quantum behavior, I'm saying there is a kind of myth about all kinds of quantum behavior that "have no classical analog", when in fact they very often have fairly obvious classical analogs, which has a lot to do with why they could have been, or were, first encountered in a classical theory. Two-slit diffraction being a prime example, the uncertainty principle being another. The "quantum weirdness" in the HUP is not at all that is has no classical analog, it is simply that its classical analog is something we didn't expect to apply to quanta. That's the quantum weirdness of it, it is that the classical analog is not the classical analog we expected, not that there is no classical analog. Just look at the language of quantum mechanics, do we have a "quantum function"? No, we have a wave function. Waves are classical concepts, so this is another perfect example of the relevance of classical analogs in what gets sold as "purely quantum" behavior.

Why not accept the limitations we find when measuring, instead of filling in the space with ideas describing notions that's already questionable, distance and motion?

This sounds like a different issue, like you are inserting an alternative interpretation to quantum mechanics. Any interpretation is fine by me as long as:
1) we don't take it too seriously,
2) it generates interesting ontological structures to ponder, and
3) it gets the predictions right.

 

[unusualname]

I don't even know what you're talking about, you're saying quantum mechanics is some kind of data-poor area of physics? That's just baloney, quantum mechanical predictions confront a spectacular amount of extremely accurate data all the time. I don't even see how this wrong point would be relevant if it were right!

I guess you see some distinction in those two sentences, but I'm afraid I do not.

QM is a theoretical model of the world which required a sensational paradigm break from classical physics, the very basic essentials were proposed by de Broglie and Heisenberg for purely theoretical reasons.

I have understood from you this week that the recent Nature article where the QM wave function is "measured" has nothing of merit that a classical analysis could not explain.

I think you are confusing the correspondence principle with something other than what it is, in particular it doesn't say that anything we measure has a classical description/explanation.

 

[Ken G]

QM is a theoretical model of the world which required a sensational paradigm break from classical physics, the very basic essentials were proposed by de Broglie and Heisenberg for purely theoretical reasons.

Yes, I believe we can all agree on that.

I have understood from you this week that the recent Nature article where the QM wave function is "measured" has nothing of merit that a classical analysis could not explain.

That is my opinion, although I laid out the map for how my claim could be falsified: simply describe a meaningful difference between the outcome of the "average trajectory" plot and a classical Poynting flux streamline diagram. No one has yet attempted to claim there is a meaningful difference in those figures, yet they seem perfectly willing to believe there is without any evidence for that belief. Simple skepticism demands that we ask for evidence of a difference, and an argument that the difference is somehow "quantum" in nature.

I think you are confusing the correspondence principle with something other than what it is, in particular it doesn't say that anything we measure has a classical description/explanation.

At no time did I ever say any such thing. What I did say is that every quantum theory spawns a classical one that describes the classical limit, I said that the classical theory could have been discovered without the quantum theory, that this often did happen in fact, and when it did not it's just an accident of history. I also said one ramification of this fact is that much of the behavior that appears in a one-at-a-time type quantum experiment has a classical analog that appears in the many-at-a-time classical limit. I also said that all this is an example of the correspondence principle.

 

[unusualname]

That is my opinion, although I laid out the map for how my claim could be falsified: simply describe a meaningful difference between the outcome of the "average trajectory" plot and a classical Poynting flux streamline diagram. No one has yet attempted to claim there is a meaningful difference in those figures, yet they seem perfectly willing to believe there is without any evidence for that belief. Simple skepticism demands that we ask for evidence of a difference, and an argument that the difference is somehow "quantum" in nature.

I was referring to http://www.nature.com/nature/journal/v474/n7350/full/nature10120.html

At no time did I ever say any such thing. What I did say is that every quantum theory spawns a classical one that describes the classical limit, I said that the classical theory could have been discovered without the quantum theory, that this often did happen in fact, and when it did not it's just an accident of history. I also said one ramification of this fact is that much of the behavior that appears in a one-at-a-time type quantum experiment has a classical analog that appears in the many-at-a-time classical limit. I also said that all this is called the correspondence principle.

Yes but the classical analog is truly misleading. In fact it turns out that the only reason classical physics works the way we once thought is precisely because the quantum "one-at-a-time" explanation is primary. There are no classical waves, they emerge from much more sophisticated QFT, that you have this backwards is almost comical.

 

[Ken G]

I was referring to http://www.nature.com/nature/journal/v474/n7350/full/nature10120.html

Oh that one, that wasn't the subject of any thread nor did I ever give it careful consideration, except to point out they could not actually be measuring the real and imaginary parts of the wavefunction independently because all wavefunctions have an arbitrary global phase. That might be an interesting thread, but note we did have a thread on the "average trajectory" paper, and I made extensive comments on that.

Yes but the classical analog is truly misleading. In fact it turns out that the only reason classical physics works the way we once thought is precisely because the quantum "one-at-a-time" explanation is primary.

How does the first sentence follow from the second? A classical analog is not misleading if you understand what it is-- a classical analog.

There are no classical waves, they emerge from much more sophisticated QFT, that you have this backwards is almost comical.

"There are no classical waves." Interesting. But you do think there are "quantum waves?" You don't think they "emerge" from some more fundamental theory too? This is just science, folks. You apparently do not really understand what I'm saying.

 

[unusualname]

"There are no classical waves." Interesting. But you do think there are "quantum waves?" You don't think they "emerge" from some more fundamental theory too? This is just science, folks. You apparently do not really understand what I'm saying.

I'll let you off the perhaps embarrassing posts you made about the Nature paper.

But this statement, wow, so you're now claiming that QM itself is not fundamental, and that helps what you've been arguing above?

QM is the WHOLE OF PHYSICS AS FAR AS WE KNOW, there is little doubt about that, the biggest effort by the most brilliant people is to make it explain gravity.

But it's pointless to mix this great intellectual endeavour up with silly discussions about the correspondence principle and interpretations.

 

[SpectraCat]

First you need a classical measurable. Electron energy flux density will suffice (we could use electron number flux, but my point is that we never need to think of these things as particles at all to "understand" diffraction). Now you need a wave theory. Huygen's principle works fine. Let's simplify life and just get a theory that works for electrons of a given energy (which here means a given ratio of energy flux to mass flux). The wave equation with the v of that population of electrons will work fine, where v is found from timing experiments. Now we need a concept of frequency because it's a wave theory, and here we can leave the frequency as a free parameter that the interference experiment will determine. The wave equation describes the speed of signal propagation, Huygen's principle tells us how to handle the sources and the slits, and the frequency parameter gives us the interference we need. Every one of these is a 100% classical concept, remember that we are pretending we don't even know we have particles here. Now we put them together to calculate the energy fluxes everywhere subject to the free frequency parameter, compare to experiments, and poof, both the frequency parameter drops out, and the fact that we have what we would call a correct theory for electron diffraction, and all classical.

This will all work fine as long as the electron ensemble is prepared in a uniform way, say by using a fixed potential drop to accelerate the beam (which in our present quantum understanding would say gives "monoenergetic electrons"). If you object that the theory so far only handles one voltage, we can just repeat the whole experience for other values of the voltage, ultimately tracing out the dependence of frequency on voltage. Still purely classical, and now we have the "Green's functions" for a complete classical description of any type of electron diffraction experiment on a large ensemble of electrons, so long as we know how that ensemble was generated. All perfectly mundane classical physics, we can do the whole thing imagining we have an "electron field" and no electrons at all.

That we can always do all this depends on only one thing: the correspondence principle. I'm saying the correspondence principle could be reframed thusly: "If the experiment is done in the classical limit, it will be describably by a classical theory", or "any quantum theory gives birth to a classical theory in the limit of large occupation numbers."

That is not an equation .. it's a phenomenological description of how might one generate an equation ... but as they say in the South, "many a slip 'twixt the cup and the lip" ... so please write down the equation.

However, based on what you write above, I get enough of the sense of what you are saying to understand that what you describe is NOT a classical theory of electron diffraction based on fundamental principles, but rather only a phenomenological model for rationalizing the observation of the experimental result of electron diffraction. If you recall, I asked for the classical equation that *predicts* the phenomenon of electron diffraction.

Finally, your use of the correspondence principle in this discussion is a red herring as far as I can tell. The fact is that you will get the same experimental results for an electron beam, whether it is firing electrons one at a time in a carefully controlled way, or as a continuous flux. So the phenomenological approach you are taking has nothing to do with "electron energy fluxes" or the use of a "large ensemble of electrons". I still want to see that equation though ...

And you trace that to its quantum nature? What in the Schroedinger equation jumps up and says "I'm a quantum"? Nothing.

If you really think that, then I guess you don't understand the Schrodinger equation very well. First of all, the Schrodinger equation is not even a wave equation. A classical wave equation equates the second time derivative of a scalar function representing the wave to the scaled Laplacian of that scalar function. The Schrodinger equation relates the FIRST time-derivative of the (scalar) wavefunction to the scaled Laplacian of the wavefunction. Furthermore, the square root of -1 appears on the RHS of the Schrodinger equation, whereas all the constants on both sides of a classical wave equation are real.

Again you seem to be saying there is no correspondence principle. But there is.
Labelling something a "trick" does not make it untrue. If a procedure spawns a classical theory, then voila, you have a classical theory. I'm sorry if you feel you got tricked in the process. The Schroedinger equation for electrons spawns a nonrelativistic classical theory simply by taking the limit of h-->0 keeping h/m a constant, all it has is a different dispersion relation from the dispersionless wave equation we are most used to seeing.

Again, you are making claims about mathematical equations .. it would be much more helpful to see the mathematical derivation. The only context in which I have studied quantum equations in the limit as h-->0 is in statistical mechanics, where it was used to derive the classical limit of the quantum mechanical partition function. If you could provide a reference which goes through the derivation you describe for the Schrodinger equation, I would like to review that. I will try to derive it myself, but I am unlikely to have time for a careful treatment of that in the near future (unless it is trivial, but I haven't seen my way to a trivial answer yet).

Speaking of time .. I am out of it .. I will address the rest of your points later.

 

[Ken G]

But it's pointless to mix this great intellectual endeavour up with silly discussions about the correspondence principle and interpretations.

Translation: you don't understand what I'm saying at all. However, I have seen many posts on this forum that suggest I am being understood by others, so I am perfectly content.

 

[unusualname]

Translation: you don't understand what I'm saying at all. However, I have seen many posts on this forum that suggest I am being understood by others, so I am perfectly content.

Yes Ken G you had (in the past) discussions in a restricted environment that seem to make sense to you and (maybe) the other participants. But you haven't had someone like me to clarify the situation and in fact point out that you aren't making sense.

 

[DevilsAvocado]

Ops... that’s a long discussion and a lot of words... but I don’t think it’s necessary to reply on every sentence, because:

The Correspondence principle only works for large quantum numbers, right?

Correct

There is no 'escape route' around this, and the only logical conclusion must then be:

The Correspondence principle does not work for (small or) single quantum numbers.​

Why is this so 'hard' to spell out...?

 

[Ken G]

If you think that obvious point is in any way relevant to what I said, it is hopeless that you will understand what I did say (which is that many quantum phenomena can be better understood by noticing their classical analogs). Indeed, I've mentioned several classical analogs in this thread, I explained how they could have been arrived at even without knowing that we are dealing with quanta (like two-slit diffraction patterns), and still you cling to the empty claim that this somehow doesn't make sense.

Maybe you can answer this: just what do you think is inherently "quantum" in a two-slit diffraction pattern? I'm all ears.

Meanwhile, I'll just have to hope that someone else was lurking in this thread, someone who actually did wish to understand the correspondence principle, and who might glean the importance of these words: every quantum theory spawns a classical theory that has to work in the classical limit. That's what "classical analog" means, for those watching at home.

 

[Ken G]

Yes Ken G you had (in the past) discussions in a restricted environment that seem to make sense to you and (maybe) the other participants. But you haven't had someone like me to clarify the situation and in fact point out that you aren't making sense.

If you think something I said didn't make sense, you are welcome to quote it again, and I'll explain it to you again. But just because you don't understand something, and completely misconstrue it every time you try to sumamrize it, doesn't mean it didn't make sense when it was actually said.

 

[Demystifier]

The charge is the property that causes the "particle" to "feel" a force in an applied electric field. In the case of electrons, it is also the property which causes it to give rise to a measurable signal when it impacts the detector. How do you define it?

I will use your definition - the cause of the force. But in Bohmian mechanics, the cause of ALL forces on particles is the wave function. Instead of dealing with various fields such as gravitational and electric field, you need only one "field" - the wave function. In this sense, there is only one force on particles - the quantum force caused by the wave function. But form such a point of view, the charge (i.e., the cause of electric force) does not have a fundamental meaning. It is a useful concept only in the classical limit of forces on particles, and for forces on the wave functions.

Can you please elaborate, I know even less about MWI than I do about BM?

It is explained in many places. See e.g.
http://en.wikipedia.org/wiki/Bohm_interpretation#Collapse_of_the_wavefunction

I guess I don't really know what you are getting at. I guess it is "special" in CM because position indexes spatial coordinates and allows us to describe the relative orientation of objects (kind of a circular description I know) ... perhaps it even "defines" space itself in some context? This is the kind of foundational question that I don't have much experience thinking about .. my background in quantum comes from the chemistry side, so my training in foundational issues, and even CM , is rather weak I am afraid. That is one of the reasons I find PF so valuable.

My point is that the "speciality" of the position observable in BM is fully analogous to that in CM. Therefore, to postpone controversy as much as possible, it is much better to first answer that question within CM, without referring to QM. For basics of CM see e.g.
http://en.wikipedia.org/wiki/Classical_mechanics

 

[SpectraCat]

First you need a classical measurable. Electron energy flux density will suffice (we could use electron number flux, but my point is that we never need to think of these things as particles at all to "understand" diffraction). Now you need a wave theory. Huygen's principle works fine. Let's simplify life and just get a theory that works for electrons of a given energy (which here means a given ratio of energy flux to mass flux). The wave equation with the v of that population of electrons will work fine, where v is found from timing experiments. Now we need a concept of frequency because it's a wave theory, and here we can leave the frequency as a free parameter that the interference experiment will determine. The wave equation describes the speed of signal propagation, Huygen's principle tells us how to handle the sources and the slits, and the frequency parameter gives us the interference we need. Every one of these is a 100% classical concept, remember that we are pretending we don't even know we have particles here. Now we put them together to calculate the energy fluxes everywhere subject to the free frequency parameter, compare to experiments, and poof, both the frequency parameter drops out, and the fact that we have what we would call a correct theory for electron diffraction, and all classical.

This will all work fine as long as the electron ensemble is prepared in a uniform way, say by using a fixed potential drop to accelerate the beam (which in our present quantum understanding would say gives "monoenergetic electrons"). If you object that the theory so far only handles one voltage, we can just repeat the whole experience for other values of the voltage, ultimately tracing out the dependence of frequency on voltage. Still purely classical, and now we have the "Green's functions" for a complete classical description of any type of electron diffraction experiment on a large ensemble of electrons, so long as we know how that ensemble was generated. All perfectly mundane classical physics, we can do the whole thing imagining we have an "electron field" and no electrons at all.

That we can always do all this depends on only one thing: the correspondence principle. I'm saying the correspondence principle could be reframed thusly: "If the experiment is done in the classical limit, it will be describably by a classical theory", or "any quantum theory gives birth to a classical theory in the limit of large occupation numbers."

And you trace that to its quantum nature? What in the Schroedinger equation jumps up and says "I'm a quantum"? Nothing.

I already responded to this section (and you have not yet responded to that post), but here I would like to emphasize the non-relevance of the correspondence principle to this whole line of discussion. As you point out, the correspondence principle deals with systems in the limit of large quantum numbers, and explains why objects with macroscopic sizes and masses can be described with arbitrary accuracy by classical mechanics. So, in the example above, what is different in terms of quantum numbers about the case you describe for a "large ensemble of electrons", and the case where electrons are sent through the double slit "one-by-one". The use of the word "ensemble" suggests that it is statistics, rather than the correspondence principle, that are relevant to your case, but of course the statistical behavior is unchanged.

On the other hand, for sufficiently high electron fluxes, there likely WILL be a difference in the experimental results, due to space charging effects. That is, the electrons will interact with each other as they pass through the slits, and that will affect the diffraction pattern. In fact, it might be that such interactions would destroy the interference pattern altogether. However in any case, a phenomenological theory to explain those results would necessarily be different from a correct quantum theory.

All of which is irrelevant to anything I've said. Note I never said "there is no such thing as a quantum effect that has no classical analog." What I did say is "many quantum effects do have classical analogs that we take advantage of all the time, especially when testing quantum mechanics, yet many people seem to be unaware of this fact."

I don't think it is irrelevant .. I raised those examples to point out that there are macroscopic systems that rely on quantum effects, which interact on the same scale with macroscopic systems where quantum effects can be ignored. The point being that taking the limit as h-->0 to get the classical limit of a theory doesn't prove anything unless you have confirmation from experiment that the result is correct. I maintain that taking the limit as h-->0 for the case of electron diffraction will give you nonsense, at least in terms of the experimentally observed results .. I have yet to see any evidence from you to the contrary.

Again, nothing we are talking about has anything to do with the philosophical interpretation that we commonly give to quantum phenomena. The point I'm making is quite a bit simpler than that: all quantum theories spawn classical theories in the classical limit, and such classical analogs can be very useful for understanding a wide array of what we call quantum phenomena, in stark contrast to what is often said about many kinds of quantum weirdness. Most quantum weirdness is simply the fact that it seems weird to us that a wave theory applies to quanta, but since wave theories are classical, we should not say that it's weird because there's no classical analog, we should say it's weird because there is a classical analog and we didn't expect that.And that just doesn't make any sense. There is no distinction between "mathematical similarities" and "physical similarities", the mathematical is all we have to understand the physical.

There is a huge difference between mathematical and physical similarities. One can write all sorts of mathematical equations that have no physical significance. The most obvious example of this is the importance of physical dimensions in physical equations ... units are irrelevant in math, while they are of paramount importance in physics. This point is relevant to the current discussion as well, since the physical dimensionality (i.e. relationships between units) of the Schrodinger equation is different from that of a classical wave equation. I already pointed this out in the context that for a classical wave, the energy is proportional to the amplitude, while in quantum mechanics, the energy is proportional to the frequency of the wave. That is not a trivial difference, and in fact is what gives rise to much of the "quantum weirdness" you mentioned.

Ah, the uncertainty principle-- another 'quantum phenomenon' with a perfectly classical analog. Comes up in Fourier transforms of classical fields all the time, thanks for bringing that up.

So what? That wasn't the point of my comment at all .. the point is that in those classical equations, it does not involve the complementarity of momentum and position, which provides the fundamental physical basis for deriving all of quantum mechanics. The correspondence principle ONLY applies in cases where the effects of the uncertainty principle can be ignored due to the large scale of the problem. That will never be true for electrons under any circumstances where electron diffraction can be observed.

 

[Ken G]

On the other hand, for sufficiently high electron fluxes, there likely WILL be a difference in the experimental results, due to space charging effects. That is, the electrons will interact with each other as they pass through the slits, and that will affect the diffraction pattern.

I'm not familiar with the term "space charging", but perhaps you are alluding to a subtle but interesting element of the correspondence principle here. Classically, the interference is in the electric fields, so there's no reference to photons and no need to worry if photons interfere with themselves or with other photons. Quantum mechanically, it seems different, because it seems like if a photon interferes with itself or another photon should be a physically significant fact. But the classical solution obeys a principle of superposition, so what is the principle of superposition a classical analog of if the individual photons think interfering with themselves is something different from interfering with each other?

Although you'll probably resist following this argument once again, perhaps even conclude I'm again embarrassing myself by trying to explain it to you, but here we have yet another perfect example of the correspondence principle and the power of classical analogs. The superposition principle is indeed the classical analog of something quantum, it's just something rather subtle that you might not know about. Photons are indistinguishable, so they don't actually have their own wavefunctions, that's just a kind of heuristic simplification we get away with if we know what we are doing. In a more complete description, there is only one wavefunction for all the photons, and the superposition principle is the classical analog of this fact.

Now, we can certainly break the wavefunction up into what are called "subspaces", and if we do that cleverly (like lumping all the photons with the same energy and spin together), we can pretend that there is one wavefunction for that subclass. But the members of that subclass have a definite phase relationship, so can interfere, and we cannot safely break up the class any further unless we intend on keeping track of that ability to interfere. That fact is just precisely what the superposition of electric fields is the classical analog of. Which gets us to the real question: what significance do you thnk this has, because it certainly doesn't support your point that there are not useful classical analogs here.

I don't think it is irrelevant .. I raised those examples to point out that there are macroscopic systems that rely on quantum effects, which interact on the same scale with macroscopic systems where quantum effects can be ignored.

Since my point is that classical analogs are useful in understanding these systems, you would of course conclude they are not if you don't understand them. So saying they are not merely underscores the point that you don't understand them, it is not a logical argument that there are no such useful classical analogs. I've given the analogs, proved their existence via the correspondence principle, shown how to get them, and explained why they are useful. All you are saying translates to "but I don't understand, so I guess they are not useful after all." Yet they are.

The point being that taking the limit as h-->0 to get the classical limit of a theory doesn't prove anything unless you have confirmation from experiment that the result is correct.

You just keep confirming that you don't understand the correspondence principle. That's all this post is-- recurrent repetition of all the different ways you don't understand that principle. I'm not sure I can explain it any better than I have, it may require someone else, or else you can just not understand that principle.

I maintain that taking the limit as h-->0 for the case of electron diffraction will give you nonsense, at least in terms of the experimentally observed results .. I have yet to see any evidence from you to the contrary.

Then you haven't been watching very carefully. I already laid out in complete detail exactly how quantum mechanics diffraction calculations spawn a purely classical version of the same theory. Since you didn't get it before, there's little point in repeating, so I'll try a different tack. You seem to think the problem is with a classical theory of electron diffraction, but I presume you have no problem with a classical theory of photon diffraction, yes? So why do you think there's a classical analog to photon diffraction, but not one to electron diffraction? Is there something special about the electron quantum that blocks the correspondence principle for that particle? How about neutrinos, will they be the kind of particle that has a correspondence principle, or the kind that doesn't?

Of course the real answer is that experiment has confirmed many times that all particles we know of satisfy a correspondence principle, so the classical theory we can make for them does in fact agree with experiments in the classical limit.

There is a huge difference between mathematical and physical similarities.

I'm sorry, that's just more nonsense, no relevance to anything being said here.

The correspondence principle ONLY applies in cases where the effects of the uncertainty principle can be ignored due to the large scale of the problem.

One more statement that exposes you just don't understand the correspondence principle. The uncertainty principle is an excellent example of the correspondence principle, if you understand the connection between the bandwidth of any classical wave and its localizability. Are you familiar with the classical notion of a Fourier transform? You might want to start your discovery of the correspondence principle there.

 

[SpectraCat]

I'm not familiar with the term "space charging", but perhaps you are alluding to a subtle but interesting element of the correspondence principle here. Classically, the interference is in the electric fields, so there's no reference to photons and no need to worry if photons interfere with themselves or with other photons. Quantum mechanically, it seems different, because it seems like if a photon interferes with itself or another photon should be a physically significant fact. But the classical solution obeys a principle of superposition, so what is the principle of superposition a classical analog of if the individual photons think interfering with themselves is something different from interfering with each other?

For crying out loud, I have been talking about ELECTRONS .. not photons! You keep confusing the two, and there are huge differences between them. Of course photons don't have a "space charging" effect, but electrons, which is what this whole side discussion is about certainly do. They are like charged particles, so if their density in a region of space is sufficiently high, then the repulsive interactions between them give rise to pertubations of the "trajectories" that are clearly not included in when space charging effects are negligible, as in the one-by-one example.

Although you'll probably resist following this argument once again, perhaps even conclude I'm again embarrassing myself by trying to explain it to you, but here we have yet another perfect example of the correspondence principle and the power of classical analogs. The superposition principle is indeed the classical analog of something quantum, it's just something rather subtle that you might not know about.

The superposition principle is neither classical nor quantum .. it is a mathematical principle that applies for any linear system.

Photons are indistinguishable, so they don't actually have their own wavefunctions, that's just a kind of heuristic simplification we get away with if we know what we are doing. In a more complete description, there is only one wavefunction for all the photons, and the superposition principle is the classical analog of this fact.

I have no idea what you are trying to say with the above ... the superposition principle has nothing to do with photons per se. It applies to the case of photons for the reason that they don't interact. As I explained above, for electrons the superposition principle does not hold if the electrons have non-negligible interactions with other charged particles. If it did, then solving electronic structure problems would be a WHOLE lot easier.

Furthermore, the superposition principle for photons does not seem completely analogous to the correspondence principle. The superposition principle preserves the phases of the waves being added, so that for coherent photons of the same frequency, the classical EM wave obtained through the superposition principle has the same phase as the individual photons that make it up. If they are not coherent, then you get interference effects that cancel some of the amplitude and shift the phase, but the phase is still well-defined. The correspondence principle is all about the "loss" of phase information (and thus coherence) as the scale of a problem becomes macroscopic. It explains why we do not observe phase effects (like interference) for macroscopic objects, although the phase information is still there at the quantum scale of course. So, if the superposition principle for photons is analogous to the correspondence principle, what information about the quantum scale is "lost" as you increase the occupation numbers of the different states of the quantum field (which is how I understand quantum numbers in relation to photons).

Now, we can certainly break the wavefunction up into what are called "subspaces", and if we do that cleverly (like lumping all the photons with the same energy and spin together), we can pretend that there is one wavefunction for that subclass. But the members of that subclass have a definite phase relationship, so can interfere, and we cannot safely break up the class any further unless we intend on keeping track of that ability to interfere. That fact is just precisely what the superposition of electric fields is the classical analog of. Which gets us to the real question: what significance do you thnk this has, because it certainly doesn't support your point that there are not useful classical analogs here.

Since my point is that classical analogs are useful in understanding these systems, you would of course conclude they are not if you don't understand them. So saying they are not merely underscores the point that you don't understand them, it is not a logical argument that there are no such useful classical analogs. I've given the analogs, proved their existence via the correspondence principle, shown how to get them, and explained why they are useful. All you are saying translates to "but I don't understand, so I guess they are not useful after all." Yet they are.

You have not done any such thing .. the discussion we have been having is for ELECTRONS, not photons. As you say, classical descriptions of electromagnetic radiation can be obtained by simply adding up photons according to the superposition principle. I am not an expert on QFT, but as I understand it, this works because photons are massless particles, and the quantum fields describing them are highly analogous to the classical descriptions of EM radiation.

None of that applies to diffraction of massive particles like electrons, which is what spawned this entire side discussion. As I said at the beginning, there is no classical theory that predicts the diffraction of electrons. You tried to argue to the contrary, but have not yet responded to my detailed criticisms of your argument, nor provided the equation that you say can be generated to describe that phenomenon based on "wave-mechanics".

You just keep confirming that you don't understand the correspondence principle. That's all this post is-- recurrent repetition of all the different ways you don't understand that principle. I'm not sure I can explain it any better than I have, it may require someone else, or else you can just not understand that principle.

I certainly understand the correspondence principle as it applies to non-relativistic quantum mechanics. I have demonstrated that throughout my comments on this thread. You have not yet supplied a convincing argument why any of my points are incorrect.

Then you haven't been watching very carefully. I already laid out in complete detail exactly how quantum mechanics diffraction calculations spawn a purely classical version of the same theory.

Not really complete detail, since you never supplied the equation that I asked for. I tried to write it from your description, but I couldn't see how to do it in the few minutes I had to devote to trying. Therefore I asked you to do it .. for whatever reason, you seem unwilling to do it. Therefore I can only conclude for now that my initial suspicion was correct, and the equation cannot be written.

Since you didn't get it before, there's little point in repeating, so I'll try a different tack. You seem to think the problem is with a classical theory of electron diffraction, but I presume you have no problem with a classical theory of photon diffraction, yes? So why do you think there's a classical analog to photon diffraction, but not one to electron diffraction? Is there something special about the electron quantum that blocks the correspondence principle for that particle?

Yes, as I have explained multiple times in detail on this thread. The correspondence principle ONLY applies in the limit where quantum uncertainty can be ignored, since electron diffraction occurs purely due to quantum uncertainty effects, it cannot be explained by any theory in the classical limit obtained according to the uncertainty principle.

As you say, that works differently for photons, and the classical description of EM-radiation turns out to predict the same results as the full quantum treatment .. I already admitted I don't completely understand why that is true, but I do know that the fact that photons have no rest mass is essential to the explanation.

How about neutrinos, will they be the kind of particle that has a correspondence principle, or the kind that doesn't?

Good question .. I believe the question of the neutrino mass is an open area of research. If they are massless, then I would expect

Of course the real answer is that experiment has confirmed many times that all particles we know of satisfy a correspondence principle, so the classical theory we can make for them does in fact agree with experiments in the classical limit.

I'm sorry, that's just more nonsense, no relevance to anything being said here.

Well, I find your comment above nonsensical (as well as somewhat insulting), because I took care to explain precisely why my comment was relevant to the context of this thread, relating it specifically to the difference in dimensionality of the Schrodinger equation and the classical wave equation. As I already commented in another post, it appears that you don't really understand the Schrodinger equation very well, at least in terms of how it relates to classical wave equations.

One more statement that exposes you just don't understand the correspondence principle. The uncertainty principle is an excellent example of the correspondence principle, if you understand the connection between the bandwidth of any classical wave and its localizability. Are you familiar with the classical notion of a Fourier transform? You might want to start your discovery of the correspondence principle there.

Wow .. now the Heisenberg uncertainty principle is an example of the correspondence principle? And you say *I* am not understanding the physics in this thread? That is really putting the cart before the horse. The correspondence principle can be perhaps be rationalized in terms of *one of* the limiting behaviors of the Fourier transform relationship between position and momentum in QM, in that in the limit of highly localized particles, the momentum distribution gets very broad, and thus includes high-k momentum states that are close to classical description. In addition, the interference between the various momentum states washes out the spatial coherence of the position-representation of the wavefunction, so it can appear highly localized in position space. However, if the mass of the quantum system is small, a highly-localized wavepacket will not remain localized, but will rapidly spread as the wavepacket propagates. Thus the full realization of the correspondence principle (i.e. localization of the particle over essentially infinite timespans) requires high mass, as well as the broad momentum distribution.

Anyway, the point is that the HUP is much more fundamental to quantum mechanics than being "just and example of the correspondence principle", as you seem to be suggesting. Furthermore, this point applies directly to the example we have been discussing, since I have pointed out several times (and you have not addressed or acknowledged), that the correspondence principle does not apply for electron diffraction, because the phenomenon requires NOT being in the limit of high quantum numbers where the quantum wavelength of the electrons can be ignored.

 

[Ken G]

For crying out loud, I have been talking about ELECTRONS .. not photons! You keep confusing the two, and there are huge differences between them.

Yes, there are interparticle forces that could affect the classical limit, but I'm talking about the simplest classical limit for understanding diffraction. One might call it "collisionless hydrodynamics". So the hydro of ideal gases. That's the "classical limit" I'm referring to for the purpose of applying the correspondence principle. It's really plasma physics, so it's an idealization because there are lots of additional plasma modes there too, all of which are classical limits of electron-proton quantum interactions. But for the purposes of analyzing the classical limit of two-slit diffraction, we can ignore those complications. Fortunately the "classical limit" sets in long before the densities get high enough to have to worry about DeBye screening and so forth, but perhaps it would be best to imagine the two-slit diffraction pattern of a quantum that was just like an electron but had no charge, and let me ask you this: do you think a particle just like an electron but with no charge has the same, or different, quantum mechanical description of its diffraction? There's not much point in the rest until you can answer this one. But this needs clarification:

Wow .. now the Heisenberg uncertainty principle is an example of the correspondence principle?

My point about the HUP was that the HUP, which is a quantum principle, has a very clear classical analog, via the correspondence principle. Now do you think that's true, or don't you?

So let's just concentrate on these three questions, they are the heart of the matter:
1) Do you think an electron with no charge would show the same diffraction pattern? Yes or no please.
2) Do you think the HUP has an obvious classical analog, which can be used to understand the HUP better, and the existence of this analog is a perfect example of the correspondence principle? Yes or no please.
3) Do you think this statement is true: "every quantum theory can be used to spawn a classical theory, that invokes only classical concepts (including no quanta), that will always work in the classical limit of large occupation numbers of that same quantum system, and what's more, it is essentially an accident of history as to which was discovered first, the quantum theory or it's classical analog." Yes or no please.

Answer these, then we can see whether what I've been saying makes sense. In contrast, what does not make sense is this statement you just made:

The correspondence principle ONLY applies in the limit where quantum uncertainty can be ignored, since electron diffraction occurs purely due to quantum uncertainty effects, it cannot be explained by any theory in the classical limit obtained according to the uncertainty principle.

Translation: you don't understand the correspondence principle. Just think about photons, please-- unless you believe that electron diffraction occurs purely due to quantum photon uncertainty effects, but photon diffraction occurs for some other reason! Is that what you think? (I guess that's question #4. You can't learn the correspondence principle without answering those 4 questions.)

One more thing: here is another false statement by you about the correspondence principle:

The correspondence principle is all about the "loss" of phase information (and thus coherence) as the scale of a problem becomes macroscopic.

I regret having to be the one to tell you, but that's just completely wrong. The correspondence principle applies perfectly well in situations where phase is preserved, all it requires is large occupation numbers. Look it up. Think about a laser beam, and Maxwell's equations.

 

[yoron]

The correspondence principle makes sense if we define our macroscopic reality as hinging on what we see from a QM perspective.

Assuming that we have a causality chain that not only develop in the arrow of time but also according to some principle of limits of observation. To create such an assumption it seems as we introduce a question of 'size' too? As if we had something not only defining a classical causality, but also at 'some right angle of thought' :) also somehow related to its 'absolute size, or relative might be a better word?', as defined from where we observe macroscopically.

A little like this perhaps, when we come down to QM the chunks of 'time' we measure shrinks, not only the 'sizes' of whatever we measure, but also its duration? So 'size' and 'time'?

You made me look it up Ken, and it was interesting reading.

"According to the selection rule interpretation, Bohr's correspondence principle is best understood as the statement that each allowed quantum transition between stationary states corresponds to one harmonic component of the classical motion. More precisely, Bohr's selection rule states that the transition from a stationary state n′ to another stationary state n″ is allowed if and only if there exists a τth harmonic in the classical motion of the electron in the initial stationary state; if there is no τth harmonic in the classical motion, then transitions between stationary states whose separation is τ are not allowed quantum mechanically. "

But Bohr seems to have changed his definition from where he first defined it as "the radiation is not a result of the accelerated motion of the electron in its orbit, but rather of the electron jumping from one stationary state to another; and rather than giving off all of the harmonic “overtones” together, only a single frequency, ν, is emitted, and the value of that frequency is given by the Bohr-Einstein frequency condition. The spectral lines are built up by a whole ensemble of atoms undergoing transitions between different stationary states, and these spectral lines, though they exhibit a pattern of regularity, are not evenly spaced—except in the limit of large quantum numbers."

This is a very interesting definition to me, especially as we seem to have defined a 'shape' to that 'electron' as perfectly round. That statement, as compared to the idea of a electron as representing a probability function until measured, collides in my head. Myself I still see it that way. and when it comes to 'classical limits', it always makes me think of the way we classically define a 'perfect space' as being null and 'empty'. Which is a correct statement to me macroscopically.

Classical limits is another definition I will need to look up.
Bohr's Correspondence Principle. I will have to reread it a couple of times, for sure :)

 

[Ken G]

Please read the rest of my post carefully before responding .. I explain in detail why the answer is no.

Then go to my question #1 and we'll start there, because charge is a complete red herring to this discussion.

 

[Ken G]

You made me look it up Ken, and it was interesting reading.

"According to the selection rule interpretation, Bohr's correspondence principle is best understood as the statement that each allowed quantum transition between stationary states corresponds to one harmonic component of the classical motion. More precisely, Bohr's selection rule states that the transition from a stationary state n′ to another stationary state n″ is allowed if and only if there exists a τth harmonic in the classical motion of the electron in the initial stationary state; if there is no τth harmonic in the classical motion, then transitions between stationary states whose separation is τ are not allowed quantum mechanically. "

That's very interesting, you can see how classically Bohr liked to think, and how deep was the intuition he gained from it. Too bad he's not posting to this thread! Your quote actually raises an aspect of the correspondence principle I had never seen-- a general and formal way to use the Fourier modes of classical wave equations to impose constraints on quantum laws. I had only ever thought of it in the other direction, that it was a way to derive classical laws from quantum mechanical ones, but of course that does imply constraints on the quantum laws if we already have the working classical theory. Fascinating, thank you.

 

[unusualname]

The correspondence principle is not a law of nature, it's simply a rule for us dumb humans to check that our (quantum mechanical) model of reality is consistent.

The correspondence principle enables us to constrain our models of the microscopic world, simply because these models must (obviously) also describe the macroscopic world we observe in appropriate limits (of large numbers)

However, the correspondence principle says nothing more than that about reality.

Why would anyone resurrect an ancient philosophical principle to attempt to explain modern physics experiments?

Nobody really cares about this out-dated way of thinking anymore, we now have the far more sophisticated decoherence argument to explain macroscopic phenomena.

 

[Ken G]

The correspondence principle is not a law of nature, it simply a rule for us dumb humans to check that our (quantum mechanical) model of reality is consistent.

I'm afraid that's a pretty good definition of a "law of nature." Why you see a distinction there is certainly outside anything that could be called science.

Why would anyone resurrect an ancient philosophical principle to attempt to explain modern physics experiments?

Yeah, like a principle of symmetry among observers. What good physics could that lead to either?

Nobody really cares about this out-dated way of thinking anymore, we now have the far more sophisticated decoherence argument to explain macroscopic phenomena.

Like I said, if you don't understand the correspondence principle, you can't use it to understand various types of quantum phenomena.

 

[unusualname]

^^Not worth replying to directly, to add something interesting, Chaos theory arguably made the correspondence principle useless:

Why We Don't Need Quantum Planetary Dynamics: Decoherence and the Correspondence Principle for Chaotic Systems

 

[SpectraCat]

Then go to my question #1 and we'll start there, because charge is a complete red herring to this discussion.

Well, I would prefer that you read in full my previous discussions of these topics, and address all the points. It's not about charge, it's about MASS. As I have now said several times, it is the wavelike behavior of MASSIVE particles that can never be obtained in the limit of large quantum numbers, because it is precisely that information that is "averaged out" according to the correspondence principle. It's amusing that you think I am the one who has a misunderstanding of the correspondence principle in this respect ... I think exactly the opposite.

Charge is not a red herring in the context in which I used it, which was to explain why you might observe deviations from the quantum mechanical limit if you used high fluxes of electrons to carry out the experiment. However, I agree that it is not relevant to the main issue.

With regard to your "questions" here are my answers:

1) Do you think an electron with no charge would show the same diffraction pattern? Yes or no please.

That is not a valid question without further caveats, so I will supply them. Assuming we could have massive particles with the same mass as the electron, AND we could create monoenergetic beams of them, AND those particles interacted with the ordinary matter of the double slit through the electromagnetic force (i.e. like a neutral atom would), then yes, I would fully expect to see diffraction.

2) Do you think the HUP has an obvious classical analog, which can be used to understand the HUP better, and the existence of this analog is a perfect example of the correspondence principle? Yes or no please.

I already answered this question as best I can in my last post ... my answer is again a qualified yes. The correspondence principle can be appreciated in the context looking at the limit of the Fourier transform of a highly localized wave packet with a broad momentum distribution. That does NOT make the HUP "an example of the correspondence principle", which is the specific statement you made that I objected to. Furthermore, the true understanding of the correspondence principle comes not from this analogy, but rather by looking at the relative magnitudes of the uncertainties as the systems approach macroscopic sizes.

3) Do you think this statement is true: "every quantum theory can be used to spawn a classical theory, that invokes only classical concepts (including no quanta), that will always work in the classical limit of large occupation numbers of that same quantum system, and what's more, it is essentially an accident of history as to which was discovered first, the quantum theory or it's classical analog." Yes or no please.

Of course I agree, that is what the correspondence principle tells us. The issue is that such an treatment CANNOT be generated to describe the diffraction of electrons. You seem to be confusing having high occupation numbers for quantum states of EM-fields (i.e. photons), with having large numbers of monoenergetic electrons. Those two situations are NOT physically analogous. As I said before, if diffraction can be observed for the electrons, then BY DEFINITION they are not in the limit of large quantum numbers. Rather, for the double slit experiment, the observation of diffraction means that the momentum of the electron is sufficiently small that their wavelength is comparable to the width and separation of the slits. Do you disagree that is a requirement for electron diffraction? Do you disagree that electrons which have that property are NOT in the large quantum number limit? Note that the meaningful quantum number in this case refers to the momentum state (or kinetic energy) of the electrons.

Just think about photons, please-- unless you believe that electron diffraction occurs purely due to quantum photon uncertainty effects, but photon diffraction occurs for some other reason! Is that what you think? (I guess that's question #4. You can't learn the correspondence principle without answering those 4 questions.)

Yes that is what I think, as I have explained at length. Of course you can make the trivial observation that the reason that photons and electrons both exhibit diffraction in the double slit is that both have wavelengths that are greater than or equal to the spacing and width of the slits. However, while there is a classical analog of photon diffraction the diffraction of classical EM fields, there is NO classical analog for electron diffraction. If you develop a phenomenological theory from experimental results to try to model and explain electron diffraction, it will necessarily be a quantum one, i.e. one that accounts for the spatial phase of massive particles, and is only valid in the limit where h does not go to zero.

In summary, I am fine with using the correspondence principle to connect diffraction phenomena of quantum photons to classical EM waves. However, as I noted previously, it seems a bit trivial to me ... the reason this works is because the quantum description of the field of the massless photons is essentially the same as the classical EM field .. the only difference is the occupation numbers of the quantum states of the field. For massive particles, the correspondence principle explains why the spatial delocalization of the quantum states can be neglected in the limit of high quantum numbers. Therefore, if you observe an experimental effect that can only be explained by the delocalized nature of quantum particles, it cannot be explained in the classical limit using the correspondence principle. I can't say it more clearly than that, and I can't believe you won't accept that distinction.

 

[yoron]

Now, this might seem simplistic but isn't all classical systems non linear, that is chaotic, when looked over a longer period of time? And the linearity we find is defined from mathematics, not nature?

Isn't it just that we always found reality to be more understandable when confined to a constricted system with clear delimitations? And that it is this chaos theory lift forward, that in open systems you will find chaotic components. To me chaos theory speaks about patterns repeating themselves, unable to backtrack to any origin although being the result of bifurcations creating a 'linear causality chain', even if not traceable coming back at certain intervals? I'm not sure I find it meaningful to put that against the correspondence principle myself?

The correspondence principle as an idea makes a lot of sense, if we want to keep causality. Chaos theory is also about a 'causality' of sorts, just not the old 'linear one'. The correspondence principle could do as well for a non linear reality, as it seems to me?
=

It's a interesting point to make unusualname. I've seen two definitions of classical non-linearity (chaos), and QM recently. One stating that QM contain no chaos. "Let us set the record straight: there is no such thing as quantum chaos. The term “quantum chaos” is a shorthand for the study of quantized systems who’s classical analog exhibits chaotic features. This raises two obvious questions: why does quantum chaos not exist and, since that is the case, why is the study of quantized chaotic systems of interest?" from "A Rough Guide to Quantum Chaos by David Poulin." And then another that seems to explore the possibility of QM containing Chaos. Myself I started to wonder about HUP, versus our classical 'chaos'? That is if you could find a correspondence between them?Quantum Chaos.

 

[Ken G]

^^Not worth replying to directly, to add something interesting, Chaos theory arguably made the correspondence principle useless:

Why We Don't Need Quantum Planetary Dynamics: Decoherence and the Correspondence Principle for Chaotic Systems

Well done digging up another interesting paper for these discussions. That paper is a good entry point into the issues of the correspondence principle as it relates to classical mechanics (Newton's laws). So far we've mostly been talking about the correspondence principle as it relates to classical wave theory, but it's nice to have the other end of the correspondence of wave/particle duality, the duality of correspondences if you will.

From what I've seen of that paper, I feel it lays out a nice foundation for analysis, but it is simply misidentifying what the correspondence principle should mean in chaotic systems-- they interpret the CP as saying that the classical theory and the quantum theory should produce the same trajectory for an initial state, but that's asking too much in view of sensitivity to initial conditions. The more important concern is that the correspondence principle should generate the "same physics" from the classical and quantum theories, which does not require the same trajectory when the trajectory itself has no real physical meaning (the "butterfly effect"). Instead, the "same physics" should simply mean the same statistical tendencies, the same probability distribution of outcomes for any realistic uncertainty in the initial conditions.

In other words, if the quantum theory said "Hyperion will hit the Earth in 2030" and the classical theory said "Hyperion will be ejected from the solar system in 2020", then we'd have a breakdown of the correspondence principle. But neither theory can make predictions like that, and if they both agree that "the chance that Hyperion will hit the Earth by 2030 is 1 part in a quintillion", then we have the correspondence principle working fine.

 

[Ken G]

Thank you for answering my questions, now we can get somewhere:

Assuming we could have massive particles with the same mass as the electron, AND we could create monoenergetic beams of them, AND those particles interacted with the ordinary matter of the double slit through the electromagnetic force (i.e. like a neutral atom would), then yes, I would fully expect to see diffraction.

OK good, we agree there. Now for the next question: do you think that large numbers of these particles passing through the apparatus, still in the collisionless limit, would also diffract and show an interference pattern?

You seem to be confusing having high occupation numbers for quantum states of EM-fields (i.e. photons), with having large numbers of monoenergetic electrons. Those two situations are NOT physically analogous.

Well, I guess my question above will get to the heart of whether or not your claim here is really true.

Rather, for the double slit experiment, the observation of diffraction means that the momentum of the electron is sufficiently small that their wavelength is comparable to the width and separation of the slits. Do you disagree that is a requirement for electron diffraction?

Obviously that is a requirement,a first-year student of classical wave mechanics knows that. It is also irrelevent, the "classical limit" here is the limit of large numbers of particles, not the limit of a large momentum for a single particle. That is the "other side" of the wave/particle duality of correspondences, more appropriate to the Hyperion paper.

Do you disagree that electrons which have that property are NOT in the large quantum number limit? Note that the meaningful quantum number in this case refers to the momentum state (or kinetic energy) of the electrons.

Here's your problem, you have the wrong correspondence in mind! Everything I've said refers, quite clearly, to the correspondence to classical wave mechanics, not the correspondence to classical particle mechanics. Yet your statements here are irrelevant to classical wave mechanics, you are talking about the particle correspondence that turns an electron into a bowling ball. So the problem is now crystal clear: you may have understood one side of the correspondence principle, and think every time you hear it that it is this side, but you have not understood there is another side to that principle, the correspondence of the wave nature of quanta to the classical wave nature of ensembles of quanta.

This disconnect is severe, and means you need to reread the entire discussion with the correct correspondence in mind. But it was a useful mistake to make-- it draws out the fact that there are these two very separate types of correspondence, a duality of correspndences, which I hadn't been on the lookout for or I might have caught it sooner. When you answer my italicized question above, you will see this other correspondence that you have been missing, and that will open the door to a whole new world of using classical analogs to help understand two-slit quantum experiments for particles with mass-- as you apparently already knew could be done with particles without mass.

 

[SpectraCat]

Man, you really pick and choose the points in my posts that you want to respond to, don't you? I would really appreciate it if you would address the other points that have gone unaddressed ... I will try to summarize them below after I have dealt with your latest post.

Thank you for answering my questions, now we can get somewhere:
OK good, we agree there. Now for the next question: do you think that large numbers of these particles passing through the apparatus, still in the collisionless limit, would also diffract and show an interference pattern?

Of course (assuming they width of the energy distribution is sufficiently narrow that they can be described as monoenergetic) ... what would that change? There is no classical limit or correspondence principle at work there .. you just have more quantum particles going through the slits, so you build up the interference pattern faster than you would have when there were fewer particles. The appearance of interference fringes still depends on the quantum delocalization of the particles due to the HUP, as I have said from the beginning. As such, it cannot be described as classical.

Now I have a question for you ... let's switch back to electrons, so we are talking about a physically real case again ... by passing large numbers of particles through, you claim we are somehow studying the system in the limit of large quantum numbers. Please explain, in as much detail as possible, which quantum numbers are different (i.e. "in the large limit") between the one-by-one case we agree is clearly quantum, and the "large numbers of electrons" that you claim somehow bring us to the limit of the correspondence principle? Until you can answer that question, we can't make any progress here.

Well, I guess my question above will get to the heart of whether or not your claim here is really true.

Well, not so much your question as my own.

Obviously that is a requirement,a first-year student of classical wave mechanics knows that. It is also irrelevent, the "classical limit" here is the limit of large numbers of particles, not the limit of a large momentum for a single particle. That is the "other side" of the wave/particle duality of correspondences, more appropriate to the Hyperion paper.Here's your problem, you have the wrong correspondence in mind! Everything I've said refers, quite clearly, to the correspondence to classical wave mechanics, not the correspondence to classical particle mechanics. Yet your statements here are irrelevant to classical wave mechanics, you are talking about the particle correspondence that turns an electron into a bowling ball.

With all due respect, that last bit is utter nonsense. As I have explained repeatedly, and you have never addressed even once (except to contradict it without explanation), there is no classical limit in which massive particles behave like waves. That you would think there is, or even could be, indicates that you have a deep-seated misunderstanding of the quantum mechanical principles that you suppose yourself to be educating others about. This gets at one of the points that you have never addressed, namely how the Schrodinger equation somehow magically morphs into a classical wave equation in the classical limit, when the two equations don't even have the same mathematical form:

Here is the Schrodinger equation:

i\hbar\frac{d}{dt}\Psi\left(\vec{r},t\right)=-\frac{\hbar^2}{2m}\nabla^2<br /> \Psi\left(\vec{r},t\right) + V\left(\vec{r}\right)\Psi\left(\vec{r},t\right)

Here is the classical wave equation:

\frac{d^2}{dt^2}u\left(\vec{r},t\right)=<br /> c^2\nabla^2u\left(\vec{r},t\right)

Note that the Schrodinger equation involves the first time derivative of the wave function, while the classical wave equation involves the second time derivative of the scalar function representing the waveform. Please explain to me how to get from one to the other in the classical limit that you claim is valid.

So the problem is now crystal clear: you may have understood one side of the correspondence principle, and think every time you hear it that it is this side, but you have not understood there is another side to that principle, the correspondence of the wave nature of quanta to the classical wave nature of ensembles of quanta.

That seems like more nonsense ... there is no classical wave nature to ensembles of quanta for massive particles. I grow tired of this .. from what I know about physics, your claim that there is some classical limit in which massive particles have wave-character is bizarre in the extreme, and is certainly not in the mainstream. Please provide a reference for the claim you are making. Normally I would have asked for it sooner, but you have shown that you know what you are talking about in other arenas, so I figured I'd eventually figure out what you are talking about here.

This disconnect is severe, and means you need to reread the entire discussion with the correct correspondence in mind. But it was a useful mistake to make-- it draws out the fact that there are these two very separate types of correspondence, a duality of correspndences, which I hadn't been on the lookout for or I might have caught it sooner. When you answer my italicized question above, you will see this other correspondence that you have been missing, and that will open the door to a whole new world of using classical analogs to help understand two-slit quantum experiments for particles with mass-- as you apparently already knew could be done with particles without mass.

Your patronizing attitude becomes tiresome. You have provided not a shred of evidence a and only vague justifications for why the correspondence you purport to exist actually does. I asked for an equation, it was not given. I provided detailed explanations of why the idea that massive particles can display wave-like properties in the classical limit makes no sense in light of the HUP (or the de Broglie equation, for that matter) .. those points remain unaddressed, except for your bizarre dismissal of the HUP as "an example of the correspondence principle". Once more I will ask you to explain why you think a phenomenon that depends exclusively on the quantum delocalization of massive particles can possibly persist in the classical limit? You have never once answered that question with a declarative statement of your thoughts on the topic .. you have only asked me more questions. I have answered your questions, now please answer mine.