Tracks in particle detectors and quantum paths

[stevendaryl]

It's very important to "get rid of the collapse" by just not introducing it. It only causes a lot of trouble, including the whole EPR debate etc. The good thing is that it's not needed at all. Instead we can simply take Born's postulate serious and take the Minimal Statistical Interpretation. That's how, in fact, quantum theory is used in practice, when real-world experiments are made in the labs and described with help of quantum theory.

In practice, there is not much difference between all the various interpretations of quantum mechanics. The Von Neuman recipe that measurement collapses the wave function works perfectly well. The various debates are really trying to understand what the quantum recipe means.

You say that in practice, you don't need anything like collapse, but I don't see that that's completely true. What you do in performing an experiment is to prepare a system in a particular state, let it evolve, the perform a measurement. But how do you prepare a system in a particular state, in the first place? Well, one approach is to use measurement: If you want to prepare electrons in the spin-up state, you start with a source of electrons, and measure the spins (via Stern-Gerlach, or whatever). Then you only use those that have spin-up. But why does measuring spin-up mean that the electron is in the spin-up state after the measurement? Isn't that a collapse-type assumption?

 

[stevendaryl]

Yes, that is part of what I'm after with my questions. Maybe not so much a detailed comparative calculation between the two cases(wich I think it's much to ask in a forum, besides my math wouldn't be up to par to seriously analyze them), but more of conceptual explanation of the key differences wrt definite trajectories between those cases.
I know the effects of the environment are tiny compared to the coulomb atraction, but I'm having difficulties seeing how that is different from the case of a particle in a chamber with say a very powerful magnetic deflecting the trajectory of the charged particle but not making it any less definite.
I'll take a look at the reference, thanks Daryl.

I'm curious: how did you know that my name was "Daryl", when my user name is stevendaryl? (Daryl is actually my middle name, and Steven is my first name, but I go by Daryl)

 

[TrickyDicky]

I'm curious: how did you know that my name was "Daryl", when my user name is stevendaryl? (Daryl is actually my middle name, and Steven is my first name, but I go by Daryl)

Didn't even notice it, I just used it for short, elsewhere I think I used Steven,

 

[atyy]

For the question of why decoherence doesn't decohere everything, the conceptual answer is that there are different strengths of interactions. From the measurement point of view, this corresponds to measuring position with different degrees of precision. For specific examples of how even in the presence of decoherence leading to non-unitary dynamics, there can be subsystems which are decoherence free and continue to evolve unitarily, one review is http://arxiv.org/abs/quant-ph/0301032 .

There's an analogous problem in the quantum Zeno effect, in which continuous accurate measurement freezes a system. However, once again, the measurement does not have to be accurate, and the system does not have to freeze completely. There can even be unitary dynamics in a subsystem. http://arxiv.org/abs/0711.4280

One of the authors on the linked papers above, Pascazio, has addressed decoherence and particle tracks in a cloud chamber. http://www.ba.infn.it/~pascazio/publications/Particle_tracks_and_the_mechanis.pdf

 

[TrickyDicky]

Thanks, atyy, I'll take a look at those references.

 

[TrickyDicky]

Ok, I can see how not having classical orbits for electrons in atoms doesn't have any bearing on having classical trajectories for electrons in cloud/bubble chambers or the Mott problem. There is a key difference in the size vs momentum imparted by environment between the tiny atom and the chamber. The "resolution" needed is much bigger for the atom and the act of measurement completely discards even the idea of a trajectory.
Thinking it over I would say that it is deceiving to think in terms of trajectories even in the case of the tracks in particle detectors. One just observes the result of interactions and constructs something like a classical trajectory.

So if one thinks about the wave function of the whole environment+original source of interactions be it alpha decay or any other process interacting with the environment, the spherical symmetry is still there and there is no paradox at all. It is only when one clings to a discontinuous view of the wave function that one gets in trouble with spherical waves vs. "linear" tracks.

Comments, criticisms?

 

[TrickyDicky]

Wave descriptions (not necessarily spherical) of individual microscopic particles may be where quantum mechanics started, but it long ago grew beyond that early formulation of the problem. Quantum mechanically two entangled particles in the singlet state are not two microscopic particles; they're a single quantum system with a single wavefunction and two sets of observables on that system. Including the environment increases the complexity of the system (enough that completely different computational methods may be needed) but even before we include it, we've lost any sense of individual microscopic particles.

(my bold)

I guess I should have payed more attention to this.

 

[Jilang]

Thinking it over I would say that it is deceiving to think in terms of trajectories even in the case of the tracks in particle detectors. One just observes the result of interactions and constructs something like a classical trajectory.

So if one thinks about the wave function of the whole environment+original source of interactions be it alpha decay or any other process interacting with the environment, the spherical symmetry is still there and there is no paradox at all. It is only when one clings to a discontinuous view of the wave function that one gets in trouble with spherical waves vs. "linear" tracks.

Comments, criticisms?

Bravo, I feel we are getting there! I am totally happy with the first paragraph, not so sure about the second paragraph though. What is this spherical symmetry idea? All backwards moving waves are self-cancelling (although actually it was not Huygens who proved this, it came later).

 

[atyy]

Ok, I can see how not having classical orbits for electrons in atoms doesn't have any bearing on having classical trajectories for electrons in cloud/bubble chambers or the Mott problem. There is a key difference in the size vs momentum imparted by environment between the tiny atom and the chamber. The "resolution" needed is much bigger for the atom and the act of measurement completely discards even the idea of a trajectory.
Thinking it over I would say that it is deceiving to think in terms of trajectories even in the case of the tracks in particle detectors. One just observes the result of interactions and constructs something like a classical trajectory.

So if one thinks about the wave function of the whole environment+original source of interactions be it alpha decay or any other process interacting with the environment, the spherical symmetry is still there and there is no paradox at all. It is only when one clings to a discontinuous view of the wave function that one gets in trouble with spherical waves vs. "linear" tracks.

Comments, criticisms?

I think the idea of whether the trajectory is continuous or not is very technical, and not very conceptual. Conceptually, one can always assume a discretization that is much finer than the spatial resolution of one's measurements, and "construct" the continuous track from these low resolution readouts. It's in the same spirit as lattice simulations of quantum field theory, which cannot have true Lorentz invariance, but can be Lorentz invariant for all practical purposes, as long as the discretization is much finer than what current experiments can see. Mathematically, it does matter, and one can worry about whether the continuum limit really exists, but I would say that at the physics "conceptual" level, there is no big distinction between continuous and discrete (there are exceptions to the rule of thumb, eg. chiral interactions on a lattice, but let's worry about that elsewhere).

In the 2 methods of analysis the conceptual puzzles are different.

1) In decoherence followed by a single measurement and collapse, the puzzle is indeed why decoherence localizes objects. The general answer is that interactions are local in space, with nearby objects interacting more strongly, and distant objects interacting more weakly. This is just a fact of nature that we incorporate into our models of decoherence. Incorporating this fact leads to localization by decoherence.

2) In the case of multiple successive measurements, it's no puzzle why localization occurs, since each position measurement will collapse the wave function into something like a definite position. The puzzle is why in many cases the track is straight enough to get a pretty accurate momentum measurement, which seems to violate the restriction on simultaneous accurate measurement of position and momentum. The answer is that in many cases, the position measurement is inaccurate. An inaccurate position measurement gives a definite position readout, but it does not localize the wave function very tightly. Once localization occurs, then given a spherical wave function, the spherical symmetry is broken on any single measurement trial, but preserved over multiple measurement trials.

 

[TrickyDicky]

Bravo, I feel we are getting there! I am totally happy with the first paragraph, not so sure about the second paragraph though. What is this spherical symmetry idea? All backwards moving waves are self-cancelling (although actually it was not Huygens who proved this, it came later).

Oh, the spherical symmetry(isotropy) idea is specific to those systems with such symmetry like it is the case for high energy physics experiments in particle colliders and bubble/cloud chambers. Not meant as something general.
It didn't occurr to me to think of the linear tracks as related to the Huygens principle but it is a nice way to get an understanding of the Mott problem, relating it to the path integral approach, for instance this quote from wikipedia seems pertinent if one trades isotropic space/medium and real waves for isotropic system wavefunction that includes the environment:

"Huygens' principle can be seen as a consequence of the isotropy of space—all directions in space are equal. Any disturbance created in a sufficiently small region of isotropic space (or in an isotropic medium) propagates from that region in all radial directions. The waves created by this disturbance, in turn, create disturbances in other regions, and so on. The superposition of all the waves results in the observed pattern of wave propagation.

Isotropy of space is fundamental to quantum electrodynamics (QED) where the wave function of any object propagates along all available unobstructed paths. When integrated along all possible paths, with a phase factor proportional to the path length, the interference of the wave-functions correctly predicts observable phenomena. Every point on wave front acts as the source of secondary wavelets that spread out in the forward direction with the same speed as the wave. The new wave front is found by constructing the surface tangent to the secondary wavelets."

 

[TrickyDicky]

I think the idea of whether the trajectory is continuous or not is very technical, and not very conceptual. Conceptually, one can always assume a discretization that is much finer than the spatial resolution of one's measurements, and "construct" the continuous track from these low resolution readouts. It's in the same spirit as lattice simulations of quantum field theory, which cannot have true Lorentz invariance, but can be Lorentz invariant for all practical purposes, as long as the discretization is much finer than what current experiments can see. Mathematically, it does matter, and one can worry about whether the continuum limit really exists, but I would say that at the physics "conceptual" level, there is no big distinction between continuous and discrete (there are exceptions to the rule of thumb, eg. chiral interactions on a lattice, but let's worry about that elsewhere).

In the 2 methods of analysis the conceptual puzzles are different.

1) In decoherence followed by a single measurement and collapse, the puzzle is indeed why decoherence localizes objects. The general answer is that interactions are local in space, with nearby objects interacting more strongly, and distant objects interacting more weakly. This is just a fact of nature that we incorporate into our models of decoherence. Incorporating this fact leads to localization by decoherence.

2) In the case of multiple successive measurements, it's no puzzle why localization occurs, since the position measurement will collapse the wave function into something like a definite position. The puzzle is why in many cases the track is straight enough to get a pretty accurate momentum measurement, which seems to violate the restriction on simultaneous accurate measurement of position and momentum. The answer is that in many cases, the position measurement is inaccurate. An inaccurate position measurement gives a definite position readout, but it does not localize the wave function very tightly.

According to the Copenhagen interpretation both methods are equivalent, or so It claims the first refence given by stevendaryl when analyzing the early works of Heisenberg and Born.

I myself see too many holes in 2) to agree they are equivalent, in any case I prefer 1), and I don't quite agree with you that there is no big distinction in this respect between a discrete and a continuous model, in fact a continuous field model easily solves the puzzle you mention for decoherence and localization for 1)

 

[Jilang]

I see some analogies here to the quantum Zeeman effect, with continuous measurement surpressing the random outcomes...

 

[TrickyDicky]

I see some analogies here to the quantum Zeeman effect, with continuous measurement surpressing the random outcomes...

You lost me here, what is the analogy?

 

[Lapidus]

How are the track leftt say by an electron in a cloud chamber and its wave function undefined trajectory related exactly?

I have not read all posts in the thread, so I'm not sure if someone gave a satisfying answer.

The uncertainty of postion-momnetum of particles is of order of the Planck constant, but the momentum of high-energy particles is of course much higher than that. So you have more "leeway" for those particles.

The same as when you squeeze a particle in tiny box, it gains higher and higher momentum in the box which in turn allows it to have higher momentum spreads according to HUP.

 

[TrickyDicky]

I have not read all posts in the thread, so I'm not sure if someone gave a satisfying answer.

The uncertainty of postion-momnetum of particles is of order of the Planck constant, but the momentum of high-energy particles is of course much higher than that. So you have more "leeway" for those particles.

The same as when you squeeze a particle in tiny box, it gains higher and higher momentum in the box which in turn allows it to have higher momentum spreads according to HUP.

Thanks, I basically got around to it in #76.

 

[atyy]

According to the Copenhagen interpretation both methods are equivalent, or so It claims the first refence given by stevendaryl when analyzing the early works of Heisenberg and Born.

I myself see too many holes in 2) to agree they are equivalent, in any case I prefer 1), and I don't quite agree with you that there is no big distinction in this respect between a discrete and a continuous model, in fact a continuous field model easily solves the puzzle you mention for decoherence and localization for 1)

What holes do you see in 2?

(I agree with the Born and Heisenberg views in the reference given by stevendaryl that both methods should be equiavlent in Copenhagen. There are some problems, since placing the Heisenberg cut requires common sense, but I don't believe the problems occur in this case. For an example of problems with wrong placement of the Heisenberg cut, see http://arxiv.org/abs/quant-ph/9712044)

I see some analogies here to the quantum Zeeman effect, with continuous measurement surpressing the random outcomes...

You lost me here, what is the analogy?

Jilang probably meant quantum Zeno effect.

 

[TrickyDicky]

What holes do you see in 2?

I generally dislike the Copenhagen interpretation, especially the artificial separation between quantum systems and classical apparatus to observe them. In this particular case of the Mott problem I find more elegant the decoherence view.

Jilang probably meant quantum Zeno effect.

Ah,ok then. I think I even mentioned the effect at the start of the thread.

 

[atyy]

I generally dislike the Copenhagen interpretation, especially the artificial separation between quantum systems and classical apparatus to observe them. In this particular case of the Mott problem I find more elegant the decoherence view.

Yes, one can take MWI and decoherence. I'm not sure MWI totally works, but let's assume it does. In that case, 2) can be rephrased as mutiple decohering events, one for each of the successive measurements. So 2) should also have an analogue in MWI.

 

[mfb]

Well, sort of. If the atoms themselves have a definite location, then interacting with the atoms would localize the particle. But why should the atoms themselves have definite locations?

Decoherence/Measurements on a timescale of picoseconds.

That's probably true, but it seems like overkill to explain a baseball.

No, this was about the atoms in the bubble chamber.

You say that in practice, you don't need anything like collapse, but I don't see that that's completely true.

All collapse-free interpretations give you a way to prepare a pure, well-known state. Even the procedure to do this in the lab is the same, just the interpretation how it happens differs.

How so?, be specific.
When I mention multiple trajectories I obviously don't mean simultaneously, but sequentially as it interacts with the chamber environment.

Sorry, how can I be more specific than "particles teleporting randomly around in a bubble chamber of arbitrary size (even with superluminal speeds if it is large enough) are unphysical"? This is contrary to all physics we know of.

I'm not sure what you are calling measurements in this context, if every interaction with the environment was a measurement in the sense of collapse then we would not see quantum effects at all, it would all be classical. Or as you say:

I never said "every interaction". Many interactions of a fast charged particle with an atom in a liquid medium count as measurement.

So if one thinks about the wave function of the whole environment+original source of interactions be it alpha decay or any other process interacting with the environment, the spherical symmetry is still there and there is no paradox at all. It is only when one clings to a discontinuous view of the wave function that one gets in trouble with spherical waves vs. "linear" tracks.

Right.
If you use the many-worlds interpretation, for example, the wave function of the whole system keeps the spherical symmetry of the initial particle. But the system quickly decomposes to many different branches that will never "see" (influence) each other again afterwards. Each branch just sees one (nearly) classical trajectory.

 

[TrickyDicky]

Yes, one can take MWI and decoherence. I'm not sure MWI totally works, but let's assume it does. In that case, 2) can be rephrased as mutiple decohering events, one for each of the successive measurements. So 2) should also have an analogue in MWI.

I don't have a single favourite interpretation but I dislike MWI even more than Copenhagen.

 

[atyy]

I don't have a single favourite interpretation but I dislike MWI even more than Copenhagen.

What interpretation are you using that has no collapse if it is not MWI?

 

[TrickyDicky]

Not sure, a strange mix of ensemble and consistent histories?

 

[StrangeCoin]

Not sure, a strange mix of ensemble and consistent histories?

As I see it there are only three options: electrons always move in continuous trajectories, sometimes, or never. Since the second one lacks logical consistency, I suppose you are investigating the possibility of the first one. But no matter how bubble chamber trajectories are compelling, you are still left with double-slit experiments and such. If you are to ever confirm those classical trajectories you have to move away from bubble chambers and grapple with those experiments that indicate otherwise, and I'm afraid there are just too many of them. Still, I'd like to see that, I never liked QM explanations myself, way too esoteric and uncomfortably paranormal.

 

[vanhees71]

That you see tracks from single particles (!) in a detector like a cloud chamber has nothing to do with the interpretation you use for quantum theory but is a well-understood phenomenon (the minimal representation is sufficient ;-)). What you see is, of course, not the particle, but a macroscopic track of the particle, due to the interactions with the gas molecules in the cloud chamber. It's a very coarse-grained picture of the particle not the particle itself!

That you see tracks as if the particle was a classical particle has been explained already very early by Mott in a famous publication

Mott, N. The Wave Mechanics of alpha-Ray Tracks. Proceedings of the Royal Society of London. Series A 126, 800 (1929), 79-84.

 

[TrickyDicky]

As I see it there are only three options: electrons always move in continuous trajectories, sometimes, or never. Since the second one lacks logical consistency, I suppose you are investigating the possibility of the first one. But no matter how bubble chamber trajectories are compelling, you are still left with double-slit experiments and such. If you are to ever confirm those classical trajectories you have to move away from bubble chambers and grapple with those experiments that indicate otherwise, and I'm afraid there are just too many of them.

Not exactly, I think you misunderstood the key point made that it is misleading to think about trajectories in all cases, then you don't have any problems either with bubble chamber tracks, electrons in atoms or double slit behaviour. It helps getting acquainted with Feynman's sum over all possible paths aproach.

Still, I'd like to see that, I never liked QM explanations myself, way too esoteric and uncomfortably paranormal.

The math formulism of QM is not esoteric or paranormal per se, certain interpretation have some of that. And in any case you should know that most of the theoretical physicists working with QM towards a quantum gravity theory beyond the Standard model naturally consider it (together with GR) as a very good approximation to the next theory and therefore incomplete as we know it.

 

[atyy]

OK, so now that we agree on the basic approaches, I have a technical question (Tricky Dicky, let me know if this is hijacking). In Mott's paper, as described by Figari and Teta's http://arxiv.org/abs/1209.2665v1 which stevendaryl linked to above, only the time-independent Schroedinger equation is considered. Why is this permitted?

I see that Figari and Teta are co-authors on an analysis that uses the full Schroedinger equation.

http://arxiv.org/abs/0907.5503
A time-dependent perturbative analysis for a quantum particle in a cloud chamber
G. Dell'Antonio, R. Figari, A. Teta
Annales Henri Poincaré
August 2010, Volume 11, Issue 3, pp 539-564

 

[TrickyDicky]

In Mott's paper, as described by Figari and Teta's http://arxiv.org/abs/1209.2665v1 which stevendaryl linked to above, only the time-independent Schroedinger equation is considered. Why is this permitted?
I see that Figari and Teta are co-authors on an analysis that uses the full Schroedinger equation.

I find this an interesting question, maybe some of the experts might give it a try. My take is that the original paper by Mott is centered on obtaining a straitgh track in the context of a spherical wave function, and for that he just has to show that in a system with an alpha-particle and two atoms the 2 atoms can only be excited if they lie in a line, so for this kind of "geometrical" solution he doesn't need to introduce any time-dependence for that function, a stationary solution is enough to show there is no problem regarding spherical vs linear.

Actually my OP was a bit beyond the specific Mott problem, it was more related to the problem of considering classical trajectories like chamber tracks(but it could equally applied to electrons trajectories in a TV CRT or rays in any vacuum tube). In all these cases the path is considered of infinitesimal width, it is not the macroscopic width of the chamber tracks or of the beam in a CRT, as it is sometimes stated to justify that the microparticle trajectory doesn't compromise the HUP.
As commented above, in these examples one either has to renounce to referring to what is observed as a trajectory or as a microparticle, whatever is psychologically less difficult, calling it both is not QM.

 

[atyy]

Actually my OP was a bit beyond the specific Mott problem, it was more related to the problem of considering classical trajectories like chamber tracks(but it could equally applied to electrons trajectories in a TV CRT or rays in any vacuum tube). In all these cases the path is considered of infinitesimal width, it is not the macroscopic width of the chamber tracks or of the beam in a CRT, as it is sometimes stated to justify that the microparticle trajectory doesn't compromise the HUP.
As commented above, in these examples one either has to renounce to referring to what is observed as a trajectory or as a microparticle, whatever is psychologically less difficult, calling it both is not QM.

In these other cases, the flight of the particle is "free". The most common use of a particle-like derivation is to show that given an initial wave function, when the final position of the particle is measured on a screen a large distance away, that final position can be used to accurately measure the initial momentum of the particle. In fact, there is a strict quantum mechanical derivation that does not involve any assumption of a classical trajectory. The basic idea is that the initial wave from a slit is Fourier transformed (momentum is the Fourier transform of position), analogous to the Fraunhofer or far field limit in classical waves.
http://www.rodenburg.org/theory/y1200.html
http://people.ucalgary.ca/~lvov/471/labs/fraunhofer.pdf
http://www.atomwave.org/rmparticle/ao%20refs/aifm%20refs%20sorted%20by%20topic/ungrouped%20papers/wigner%20function/PFK97.pdf

Nonetheless, a classical derivation with trajectories works. This is strictly correct, even from the quantum mechanical point of view, if the initial wave function is Gaussian. This is because the Wigner function, which is the quantum analogue of the classical joint distribution for momentum and position, is positive for Gaussian wave functions and can be interpreted as a classical probability distribution. Furthermore, the Schroedinger equation for a free Gaussian wave function leads to the classical Liouville equation for the Wigner function. So in this special case of a free Gaussian wave packet, even without a Bohmian interpretation, Copenhagen does permit classical trajectories.

I believe it is a matter of luck that the quantum formula remains the same, whether or not the initial wave packet is Gaussian. So I believe that for non-Gaussian wave packets, a strictly correct derivation does not involve trajectories. I think this luck is analogous to that in Rutherford scattering, where classical and quantum derivations give the same formula for inverse squared potentials.

 

[TrickyDicky]

The point remains that you cannot identify a wave packet with a particle in QM.

 

[atyy]

A wave packet is identified with a particle in QM.

 

[TrickyDicky]

A wave packet is identified with a particle in QM.

Hmmm, so what was Born's discrepancy with Schrodinger about wave packets?

 

[Nugatory]

Hmmm, so what was Born's discrepancy with Schrodinger about wave packets?

The modern understanding of "particles", based on QFT, still hadn't been hashed out. They aren't around to ask, but it's likely that they would have found common ground there.

(and it's important to remember that the straightest path to a clean formulation of a theory is almost never the historical route by which the theory was first reached. It's a lot easier to chart a course when you already know where your destination is).

 

[atyy]

Hmmm, so what was Born's discrepancy with Schrodinger about wave packets?

Here's the history according to Wikipedia http://en.wikipedia.org/wiki/Schrödinger_equation#Historical_background_and_development

According to Born, the square of the wave function gives the probability of a particle's position. That's the Born rule that's made it into canonical quantum mechanics.

So a wave packet does represent a particle, except that it does not have a definite position and momentum at all times.

 

[TrickyDicky]

So a wave packet does represent a particle, except that it does not have a definite position and momentum at all times.

Ok, doesn't that mean it can't have a classical trajectory?

 

[WannabeNewton]

A wave packet is identified with a particle in QM.

Where in QM is a wave packet identified with a particle? It is well known that such an interpretation is highly limited and as such rather useless beyond visualization. Not only is such an interpretation restricted to single-particle systems, but also it only holds for those systems wherein the wave-packet does not spread on time scales comparable to the time evolution under Schrodinger's equation so it will work for the harmonic oscillator but not for the free particle.