Tracks in particle detectors and quantum paths

[TrickyDicky]

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

 

[Vanadium 50]

Related exactly? Pretty much the same way they are related approximately.

This is a terribly vague question. You'll need to be more specific if you want any hope of getting the answer you are looking for. If it helps, the laws of quantum mechanics apply to baseballs and planets too.

 

[CWatters]

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

Very different scales.

 

[TrickyDicky]

Related exactly? Pretty much the same way they are related approximately.

This is a terribly vague question. You'll need to be more specific if you want any hope of getting the answer you are looking for. If it helps, the laws of quantum mechanics apply to baseballs and planets too.

Hi V50, I was specifically concerned about the quantum picture of particles such as electrons not having a defined trajectory, i.e. double slit settings(thus my placing the question in the QM subforum) in contrast with the clear path followed by an electron as shown in the tracks left in cloud chambers. What is the commonly given explanation to this apparent contrast?

 

[TrickyDicky]

I was interested in a discussion about collapse, the measurement problem or even the quantum Zeno effect, but I can see that it is going to be mighty hard here in general physics.

 

[stevendaryl]

Hi V50, I was specifically concerned about the quantum picture of particles such as electrons not having a defined trajectory, i.e. double slit settings(thus my placing the question in the QM subforum) in contrast with the clear path followed by an electron as shown in the tracks left in cloud chambers. What is the commonly given explanation to this apparent contrast?

The question of why linear tracks appear in cloud chambers is known as the Mott problem, named after the physicist who first investigated it in the 1920s. I found two discussions of it:
http://arxiv.org/pdf/1209.2665.pdf
http://en.wikipedia.org/wiki/Mott_problem

 

[jtbell]

Moved to Quantum Physics.

 

[TrickyDicky]

The question of why linear tracks appear in cloud chambers is known as the Mott problem, named after the physicist who first investigated it in the 1920s. I found two discussions of it:
http://arxiv.org/pdf/1209.2665.pdf
http://en.wikipedia.org/wiki/Mott_problem

Thanks for the pertinent reference to the Mott problem, I very vaguely remember having read about it but had completely forgotten it. I'll take some time to read the references through.

 

[ZapperZ]

Hi V50, I was specifically concerned about the quantum picture of particles such as electrons not having a defined trajectory, i.e. double slit settings(thus my placing the question in the QM subforum) in contrast with the clear path followed by an electron as shown in the tracks left in cloud chambers. What is the commonly given explanation to this apparent contrast?

This is a terrible concept. There are large number of situations where the classical picture of electron trajectory works! Look at the description we use to describe the beam physics for particle accelerators! They are all classical! Beam physics codes that we use, such as PAMELA, to track electron beams all consider them to be classical particles. The electron analyzers that are used to measure and detect photoelectrons, all considered these electrons having classical trajectories from the emitting surface all the way to the CCD plate!

Why would the description for the tracks in such particle detectors be any different?

Zz.

 

[stevendaryl]

This is a terrible concept. There are large number of situations where the classical picture of electron trajectory works! Look at the description we use to describe the beam physics for particle accelerators! They are all classical! Beam physics codes that we use, such as PAMELA, to track electron beams all consider them to be classical particles. The electron analyzers that are used to measure and detect photoelectrons, all considered these electrons having classical trajectories from the emitting surface all the way to the CCD plate!

Why would the description for the tracks in such particle detectors be any different?

Zz.

Well, let me quote Bohr's paraphrase of Einstein's statement of the problem:

Mr. Einstein has considered the following problem: A radioactive sample emits α-particles in all directions; these are made visible by the method of the Wilson cloud chamber. Now, if one associates a spherical wave with each emission process, how can one understand that the track of each α-particle appears as a (very nearly) straight line? In other words: how can the corpuscular character of the phenomenon be reconciled here with the representation by waves?

 

[ZapperZ]

I'm not sure I understand the point in relation to what I wrote.

A laser makes a very predictable path for the photons in its beam. Do you represent that with "spherical waves"? Why spherical waves? Why not plane waves as we would normally consider?

Secondly, I don't see how this has anything to do with the "wavefunction" that we use to describe a quantum system.

Zz.

 

[stevendaryl]

I'm not sure I understand the point in relation to what I wrote.

A laser makes a very predictable path for the photons in its beam. Do you represent that with "spherical waves"? Why spherical waves? Why not plane waves as we would normally consider?

Secondly, I don't see how this has anything to do with the "wavefunction" that we use to describe a quantum system.

Zz.

I thought the original question was about tracks in a cloud chamber, which is what the Bohr quote is about.

 

[ZapperZ]

I thought the original question was about tracks in a cloud chamber, which is what the Bohr quote is about.

It was, but you quoted that in relation to my response. I don't know what the point is in that context. Bohr's quote offered no insight into how to treat this problem other than rephrasing what Einstein has said.

It doesn't change the FACT that we do treat electrons as classical particles in many instances, with sufficient accuracy that the very device that we use to study elementary/high energy physics particles (particle colliders), were modeled with classical trajectories.

Zz.

 

[stevendaryl]

It was, but you quoted that in relation to my response.

I was trying to get back to what the actual puzzle was. I was trying to say that I think your response didn't actually address it.

I don't know what the point is in that context. Bohr's quote offered no insight into how to treat this problem other than rephrasing what Einstein has said.

It's a statement of what the puzzle is, it's not a statement of the solution. Your remarks, bringing up lasers, for instance, don't seem relevant. The explanation for why lasers seem to have definite tracks doesn't apply to particles in a cloud chamber. Or at least, I don't see any connection.

Once again, quoting one of the original researchers, Mott:

It is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space.

 

[ZapperZ]

I was trying to get back to what the actual puzzle was. I was trying to say that I think your response didn't actually address it.

Why not? If I can model electron trajectory classically in an accelerator, what's the difference with doing it in a detector or cloud chamber? It is the same thing.

It's a statement of what the puzzle is, it's not a statement of the solution. Your remarks, bringing up lasers, for instance, don't seem relevant. The explanation for why lasers seem to have definite tracks doesn't apply to particles in a cloud chamber. Or at least, I don't see any connection.

Once again, quoting one of the original researchers, Mott:

I brought up lasers because the statement you quoted brought up "spherical waves", which was odd if that is the ONLY way to describe things. I brought up the instance where we DO have waves, i.e. light, in a laser, and we can still describe its trajectory very well when we have plane waves. So the laser was a counter example of a "wave" with definite trajectory. Waves do not always have to be spherical where the path diverges.

Take note that if the particles that we detect at the detectors can't be modeled classically as far as its trajectories are concerned, the whole concept of path reconstruction that is so common in elementary particle physics experiments can be thrown out of the window.

Zz.

 

[stevendaryl]

It's a statement of what the puzzle is, it's not a statement of the solution.

The link that I gave describes how the problem has been tackled
http://arxiv.org/pdf/1209.2665.pdf

To quote from the conclusion:

As it was pointed out before, according to Born and Heisenberg, it is definitely equivalent to consider the atom of the vapor as a (classical) measurement device (case a) or as a part of the quantum system to be described by Schrödinger dynamics (case b). Such a position is meant to guarantee the consistency of the standard interpretation of quantum theory. In particular, the authors are interested in stressing the unavoidable role of wave packet reduction as the crucial rule ensuring the right correspondence between theory and observed physical world. From a purely foundational point of view their reasoning aims to make the axiomatic scheme work in any chosen way to address the cloud chamber dynamical problem and, as such, it has been adopted and shared by the majority of the physics community for a long time. However, from the concrete point of view of the physical description of quantum systems, the equivalence of the two approaches a) and b) seems difficult to be maintained. In fact, one may find hard to accept the claim that an atom of the vapor is a classical measurement device of the position of the α-particle. After all, the atom is a microscopic system on the same ground of the α-particle and there is no a priori reason to regard it as a classical system. Agreeing with this point of view, one should concede that approach b) is surely more natural. In particular, it has the important advantage to allow a quantitative analysis taking into account explicitly the physical parameters characterizing system and environment. It is only on the basis of such a quantitative investigation that it is possible to clarify the conditions under which the interaction with the environment produces the appearance of classical trajectories.

 

[stevendaryl]

Why not? If I can model electron trajectory classically in an accelerator, what's the difference with doing it in a detector or cloud chamber? It is the same thing.

Well, there might be a similar question as to why it's possible to model electron trajectories classically in an accelerator. So maybe they are related.

I brought up lasers because the statement you quoted brought up "spherical waves", which was odd if that is the ONLY way to describe things.

The point is that, at least with a simplified model of alpha decay, the alpha particle leaves the nucleus in a spherical wave. Nobody was suggesting that ALL problems are spherically symmetric.

I brought up the instance where we DO have waves, i.e. light, in a laser, and we can still describe its trajectory very well when we have plane waves.

The Mott problem is about reconciling a spherically symmetric model with outcomes that very much are not spherically symmetric. I just don't see the relevance of bringing up lasers. It's as if I said "How can a needle float on water, when it's made of steel?" and you answered by saying "Corks float, too, and they're not made of steel."

Take note that if the particles that we detect at the detectors can't be modeled classically as far as its trajectories are concerned, the whole concept of path reconstruction that is so common in elementary particle physics experiments can be thrown out of the window

I don't think that there is any doubt about the use of particle tracks. The question is how to reconcile the observations of tracks and definite trajectories with the theory, which does not appear to have definite trajectories.

 

[atyy]

The problem can be treated as continuous approximate measurement of a quantum wave function.

http://arxiv.org/abs/math-ph/0512069
A Dynamical Theory of Quantum Measurement and Spontaneous Localization
V. P. Belavkin

We develop a rigorous treatment of discontinuous stochastic unitary evolution for a system of quantum particles that interacts singularly with quantum "bubbles" at random instants of time. This model of a "cloud chamber" allows to watch and follow with a quantum particle along the trajectory in the cloud chamber by sequential unsharp localization of spontaneous scatterings of the bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained as the result of quantum filtering theory, i.e., a theory describing the conditioning of the a priori quantum state by the measurement data. We show that in the case of indistinguishable particles the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the nonstochastic Schroedinger equation only in the mean field approximation, whereas the central limit yields Gaussian mixing fluctuations described by stochastic reduction equations of diffusive type.Side note: Why can a single free particle often be treated as if there is a classical distribution of trajectories, without resorting to a Bohmian interpretation? In general this is not possible. The closest thing to a classical probability distribution is the Wigner function, but it is not a probability distribution because it is not positive. However, for a Gaussian wave function, the Wigner function is positive, and if the particle is free, the time evolution of the function is the same as the classical Liouville equation. This is why a single free particle can often be treated as if there is a classical distribution of trajectories.

Furthermore, even when the wave function is not Gaussian, although it is wrong to consider classical trajectories, the fully quantum derivation and the classical derivation sometimes lead to the same formula. This should be considered a lucky accident, just like Rutherford scattering for an inverse-square potential gives the same results whether it is treated classically or quantum mechanically.

 

[StrangeCoin]

This is a terrible concept. There are large number of situations where the classical picture of electron trajectory works! Look at the description we use to describe the beam physics for particle accelerators! They are all classical! Beam physics codes that we use, such as PAMELA, to track electron beams all consider them to be classical particles. The electron analyzers that are used to measure and detect photoelectrons, all considered these electrons having classical trajectories from the emitting surface all the way to the CCD plate!

Why would the description for the tracks in such particle detectors be any different?

Zz.

So for example in bubble chambers, is there any reason to believe the electron is not exactly in the center of those bubbles and not travailing over continuous path those bubbles describe?

Every time we can measure it we see defined continuous trajectories, every time we can not measure it, we assume it's doing something else. Inability to measure with some desired precision is usually referred to as "margin of error", it's a property of measuring tools, not really a property of what is being measured. Why then in QM this "margin of error" is considered to be an actual property of what is being measured, rather than just a consequence of inadequate measuring tools?

 

[ZapperZ]

So for example in bubble chambers, is there any reason to believe the electron is not exactly in the center of those bubbles and not travailing over continuous path those bubbles describe?

Every time we can measure it we see defined continuous trajectories, every time we can not measure it, we assume it's doing something else. Inability to measure with some desired precision is usually referred to as "margin of error", it's a property of measuring tools, not really a property of what is being measured. Why then in QM this "margin of error" is considered to be an actual property of what is being measured, rather than just a consequence of inadequate measuring tools?

I have no idea what you are talking about.

If you are referring to instrumentation uncertainty versus the Heisenberg uncertainty, this has been discussed and described numerous times already on here. They are not the same thing.

Otherwise, I see no relevance with the topic of this thread.

Zz.

 

[TrickyDicky]

The problem has been now well formulated by Steven. About the apparently preferred solution since it was proposed by Mott of introducing the environment in the analysis to solve the paradox, I have some doubts. The way I see it it is only a superficial solution in the sense that it doesn't address the bottom of the problem, for instance it doesn't really give a hint towards solving the measurement problem, it doesn't explain collapse,and it would seem this is really behind the Mott problem.
Besides, saying that integrating the environment clarifies why we observe classical linear tracks intead of what it would be expected in theory from spherically symmetric wavefunctions might give some insight but it basically defeats the concept of quantum particles as individual microscopic objects related to spherical wavefunctions, which is the starting point of the quantum theory.

FWIW the problem is IMO intimately related with the unitarity-non unitarity problem that is being discussed in a parallel thread.

 

[Nugatory]

The way I see it it is only a superficial solution in the sense that it doesn't address the bottom of the problem, for instance it doesn't really give a hint towards solving the measurement problem, it doesn't explain collapse,and it would seem this is really behind the Mott problem.

"Superficial" is somewhat in the eye of the beholder. If you don't demand more of QM than that it provide statistical predictions of the results of measurements, the Mott solution is altogether satisfactory and even provides a great deal of insight into how the microscopic and the macroscopic worlds are connected in the formalism of QM. It's not at all clear to me that the formalism of QM is the place to be looking if you want anything more.

Besides, saying that integrating the environment clarifies why we observe classical linear tracks intead of what it would be expected in theory from spherically symmetric wavefunctions might give some insight but it basically defeats the concept of quantum particles as individual microscopic objects related to spherical wavefunctions, which is the starting point of the quantum theory.

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.

 

[atyy]

The question of why linear tracks appear in cloud chambers is known as the Mott problem, named after the physicist who first investigated it in the 1920s. I found two discussions of it:
http://arxiv.org/pdf/1209.2665.pdf
http://en.wikipedia.org/wiki/Mott_problem

The problem can be treated as continuous approximate measurement of a quantum wave function.

http://arxiv.org/abs/math-ph/0512069
A Dynamical Theory of Quantum Measurement and Spontaneous Localization
V. P. Belavkin

It's interesting to compare these two approaches. The review by Figari and Teta http://arxiv.org/abs/1209.2665v1 says that conceptually Born and Heisenberg thought the problem could be treated in two equivalent ways. One in which (a) the molecules are considered to measure the system causing wave function collapse, the other (b) in which the molecules decohere the system. Mott's approach and updates like Dell'Antonio, Figari and Testa's http://arxiv.org/abs/0907.5503 and Blasi, Pascazio and Takagi's http://www.ba.infn.it/~pascazio/publications/Particle_tracks_and_the_mechanis.pdf seem to follow (b), while Belavkin's http://arxiv.org/abs/math-ph/0512069 seems to follow (a).

 

[Jilang]

I don't think any of the above answers have really addressed the question as asked. The cloud chamber relies on a series of interactions which cause ionisation. These events are frequent in the path of the particle and constitute a series of measurements. The trajectory is only undefined between those interactions.

 

[atyy]

I don't think any of the above answers have really addressed the question as asked. The cloud chamber relies on a series of interactions which cause ionisation. These events are frequent in the path of the particle and constitute a series of measurements. The trajectory is only undefined between those interactions.

My understanding is that that is essentially Belavkin's approach.

 

[StrangeCoin]

If you are referring to instrumentation uncertainty versus the Heisenberg uncertainty, this has been discussed and described numerous times already on here. They are not the same thing.

I am referring to instrumentation uncertainty being misinterpreted as an actual property of what is being measured. I believe that's exactly what the OP question is about. You don't say electron position is undefined because you measured that to be true, but because you couldn't measure any better. Isn't that right?

So now on one hand we have the thing we can actually measure, the bubble chamber trajectories, which are apparently continuous and very well defined. And on the other hand we have QM theory which wants us to believe the opposite, that electron is actually doing something else, but it sneakily only does so when we are not looking. Apart from being funny, is there any actual reason to believe this theory based on unsuccessful measurements, rather than to believe what is obvious from those measurements that were successful?

 

[TrickyDicky]

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.

Yes, that might be so and we could debate what that loss means for the theory, but when I say I'm not convinced by the explanatory power of introducing the environment to explain the classical behaviour of a quantum system I'm referring for instance to the fact that if we apply this same logic to electrons in an atom we find serious problems not to conclude that this electrons should also follow classic trajectories within the atom and they probably should crash into the nucleus they orbit. We don't because we know well that the uncertainty principle forbids it. I'm not sure exactly how the HUP is avoided in the solution of the Mott problem.

The cloud chamber relies on a series of interactions which cause ionisation. These events are frequent in the path of the particle and constitute a series of measurements. The trajectory is only undefined between those interactions.

Here what happens "between interactions" is such a key factor wrt the trajectory that is actually measured as the outcome of those interactions compared to what the theory implies that nothing is really explained by referring to it with "only undefined".

 

[atyy]

The track in a cloud chamber is not a precise measurement of position. It consists of successive imprecise position measurements. Heuristically, this is why the track can be considered a precise momentum measurement.

As for what happens in between these imprecise measurements, the time between them can be taken to zero, such that the process describes continuous imprecise measurement.

 

[Nugatory]

I'm referring for instance to the fact that if we apply this same logic to electrons in an atom we find serious problems not to conclude that this electrons should also follow classic trajectories within the atom and they probably should crash into the nucleus they orbit. We don't because we know well that the uncertainty principle forbids it.

What does the uncertainty principle have to do with the observation that electrons do not "crash into the nucleus"? And how does applying "this same logic" about including the environment in the Hamiltonian of a bound electron lead to the conclusion that the electron should follow a classical trajectory?

These are questions not arguments, because I honestly don't understand what you're saying well enough to argue with it.

 

[TrickyDicky]

The track in a cloud chamber is not a precise measurement of position.

The track describes the classical trajectory of a quantum system, this is what Mott, Heisenberg, Born and Darwin as explained in the rerence provided by stevendaryl were trying to explain within the quantum formalism. A classical trajectory has well defined momentum and position, that is what's classical about it and what seems to clash with quantum theory in the context of the yet unsolved measurement problem.

It consists of successive imprecise position measurements.

Yes, that's the premise, Mott et al. are trying to explain, how do we go from individual imprecise measurements that should lead to not definite trajectories to a classical trajectory. The arguments used by these authors: the classical detectors and the environment. If they are valid it seems like they could be also applied in principle to situations where we don't observe clasical trajectories, like electrons in an atom, but here it is not used basically on the grounds that the HUP forbids well defined orbits(see for instance PF physics FAQ footnote on why electrons don't crash into the nucleus).

Heuristically, this is why the track can be considered a precise momentum measurement.

Too heuristic, as I said in as much as it is considered a classical trajectory the track can be considered to have position as precisely measured as momentum.