Tracks in particle detectors and quantum paths

[atyy]

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 under the Schrodinger equation so it will work for the harmonic oscillator but not for the free particle.

ψ(x₁) is identified with 1 particle.

ψ(x₁,x₂) is identified with 2 particles.

ψ(x₁,x₂,x₃) is identified with 3 particles.

 

[WannabeNewton]

ψ(x₁) is identified with a particle.

ψ(x₁,x₂) is identified with two particles.

ψ(x₁,x₂,x₃) is identified with 3 particles.

A wave-packet is a Gaussian wave-form propagating through configuration space. The wave-function is a much more general concept and the wave-function of a multi-particle system certainly cannot be identified with a configuration space wave-form since the wave-function of such a system lives in a higher dimensional space.

This is exactly why TrickyDicky referred to the history behind Born's interpretation of the wave-function in light of Schrodinger's incorrect interpretation of the wave-function as a wave-packet representing a particle in configuration space.

 

[atyy]

A wave-packet is a Gaussian wave-form propagating through configuration space. The wave-function is a much more general concept and the wave-function of a multi-particle system certainly cannot be identified with a configuration space wave-form since the wave-function of such a system lives in a higher dimensional space.

OK, if that's what you mean by wave packet, I agree (I thought "wave packet" was just another way to say " wave function"). You can see my comments on Gaussian wave functions several posts up. There, in the free particle case, we can even associate classical trajectories with the Gaussian wave function.

 

[atyy]

This is exactly why TrickyDicky referred to the history behind Born's interpretation of the wave-function in light of Schrodinger's incorrect interpretation of the wave-function as a wave-packet representing a particle in configuration space.

But his post #99 replied to my post #98, where his use of the term "wave packet" would make more sense if it referred to what I called the "wave function". It is clear in my post #98 that "wave function" and "wave packet" are meant to be the same thing.

 

[WannabeNewton]

There, in the free particle case, we can even associate classical trajectories with the Gaussian wave function.

Thank you. Do you have any further reading on that?

 

[atyy]

Thank you. Do you have any further reading on that?

As far as I know, it only works for single particle free Gaussian wave functions. The idea is that the Wigner function is the quantum analogue of the classical joint probability of position and momentum. But it is not the same because in general, the Wigner function has negative bits, whereas a classical probability distribution is positive. Also, the quantum time evolution is derived from Schroedinger's equation, whereas we need the Liouville equation for the classical case. In the special case of a Gaussian wave function, the Wigner function is positive. The free particle evolution preserves Gaussianity, and surprisingly (to me) also results in the quantum evolution being the same as the classical Liouville equation. So in this special case, we can have trajectories even in Copenhagen, without a Bohmian interpretation. There's an explanation somewhere in Ganguly's essay http://dspace.mit.edu/bitstream/handle/1721.1/49800/50586846.pdf .

However, the derivation of momentum measurements from position measurements of single particles at far field using a single slit set up hold more generally (ie. even if the initial wave function is not Gaussian). Essentially, this is because the far field wave function is the Fourier transform of the initial wave function, analogous to the Fraunhofer limit for classical waves. http://www.atomwave.org/rmparticle/ao%20refs/aifm%20refs%20sorted%20by%20topic/ungrouped%20papers/wigner%20function/PFK97.pdf

 

[TrickyDicky]

But his post #99 replied to my post #98, where his use of the term "wave packet" would make more sense if it referred to what I called the "wave function". It is clear in my post #98 that "wave function" and "wave packet" are meant to be the same thing.

It wasn't so clear to me, so I referred to a wave packet explicitly.

Still I'm not able to conclude from your explanations or your references that single particle free Gaussian wave functions have classical trajectories.
"Free particles" are known not to exist in the quantum world in any case, they are just practical idealizations.

 

[atyy]

It wasn't so clear to me, so I referred to a wave packet explicitly.

Still I'm not able to conclude from your explanations or your references that single particle free Gaussian wave functions have classical trajectories.
"Free particles" are known not to exist in the quantum world in any case, they are just practical idealizations.

OK, so everywhere that I say "wave packet", I mean "wave function" (I didn't know there was a difference till now).

Free particles don't exist, so this is just an approximation. However, as long as we are just doing quantum mechanics with a fixed number of particles, these two cases in which classical trajectories seem to have some meaning are treated differently. In the Mott cloud chamber case we have decoherence throughout or multiple measurements, whereas in the case of momentum measurement from the flight of a free particle, we only have decoherence at the end of the path, or a single measurement of position at the far field location. That these are approximations ultimately mean that neither position nor momentum are perfectly accurately measured (in fact, in quantum field theory, there isn't a relativistic position operator), but they are good enough.

 

[atyy]

BTW, there is a very interesting discussion about "free particles" in this thread https://www.physicsforums.com/showthread.php?t=749257 .

 

[mfb]

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

Right, but the deviations from a classical trajectory can be negligible. The result is a trajectory that looks classical.

 

[TrickyDicky]

Free particles don't exist, so this is just an approximation. However, as long as we are just doing quantum mechanics with a fixed number of particles, these two cases in which classical trajectories seem to have some meaning are treated differently. In the Mott cloud chamber case we have decoherence throughout or multiple measurements, whereas in the case of momentum measurement from the flight of a free particle, we only have decoherence at the end of the path, or a single measurement of position at the far field location. That these are approximations ultimately mean that neither position nor momentum are perfectly accurately measured (in fact, in quantum field theory, there isn't a relativistic position operator), but they are good enough.

Right, but the deviations from a classical trajectory can be negligible. The result is a trajectory that looks classical.

We had all agreed that an approximation to a classical trajectory is possible and it is good enough in practice, still that approximation is not a quantum microparticle's classical trajectory(and if it were it wouldn't be the trajectory of a quantum microparticle) in the rigorous sense I referred to in #97 last sentence.

I think it is important to remember here that even in classical mechanics a classical trajectory is based on the idealization of extended bodies as point-like particles, in that sense it is not possible to exactly measure classical trajectories in practice either, they are just possible in the theory. In QM in the case of elementary particles they are actually considered point particles, so no true classical trajectory is possible not only in practical measurable terms but also in principle theoretically due to the uncertainty relations, that's why even in theory only approximations to a classical trajectory are allowed in QM.
The Wigner quasiprobabilistic distribution is no different, in fact it is an approximation to a classical probabilistic distribution and to use wikipedia words only "vestiges of local trajectories are normally barely discernible in the evolution of the Wigner distribution function".

 

[mfb]

What is a "quantum microparticle" - or what is not one?
You can neglect QM in the same way you can neglect the influence of gravity on particles in the bubble chamber - it is there, but you just don't (have to) care.

Nonrelativistic QM itself is just an approximation of QFT, and that might be an approximation of some more fundamental theory. So what? Does that change our view on the bubble chamber in any way?

 

[WannabeNewton]

We had all agreed that an approximation to a classical trajectory is possible and it is good enough in practice, still that approximation is not a quantum microparticle's classical trajectory(and if it were it wouldn't be the trajectory of a quantum microparticle) in the rigorous sense I referred to in #97 last sentence.

Here's a question for you: on what length scales and time scales is the classical trajectory of the particle in the bubble chamber being realized?

 

[WannabeNewton]

There's an explanation somewhere in Ganguly's essay http://dspace.mit.edu/bitstream/handle/1721.1/49800/50586846.pdf .

Brilliant, thank you. Chapter 4 was quite lucid.

 

[TrickyDicky]

Here's a question for you: on what length scales and time scales is the classical trajectory of the particle in the bubble chamber being realized?

It was already discussed previously that length scales of the tracks (or the beams in a CRT) are big enough so that no problem with the HUP ever arises, the particles have room to be sufficiently blurred to avoid it.

So it is impossible both theoretically(taking Planck scale as a theoretic limit) or in practice(much bigger limit with present technology) to realize or probe a true classical trajectory.

 

[TrickyDicky]

What is a "quantum microparticle" - or what is not one?

This is a very good question that probably is behind the OP in part. More specifically, is the concept itself helpful or is it hampering further developements of a more fundamental theory?

You can neglect QM in the same way you can neglect the influence of gravity on particles in the bubble chamber - it is there, but you just don't (have to) care.

I'm not sure what you mean by "you can neglect QM" here. What is observed in particle detectors must be in principle compatible with the Standard model if one is using it to explain it. And ultimately with QFT and QM as that's what the SM is based on.
Or are you making a point about the difference between QFT and QM?

Nonrelativistic QM itself is just an approximation of QFT, and that might be an approximation of some more fundamental theory. So what? Does that change our view on the bubble chamber in any way?

See above.

 

[WannabeNewton]

So it is impossible both theoretically(taking Planck scale as a theoretic limit) or in practice(much bigger limit with present technology) to realize or probe a true classical trajectory.

That's because "true classical trajectories" don't exist. But how is that at arms with the situation observed in the bubble chamber? Therein the ostensible classical trajectory arises thanks to the time scales and length scales involved; certainly the particle in question is not traversing a classical trajectory but given the characteristic scales we are interested in, with regards to the bubble chamber, we can use a coarse-grained picture to describe the particle as moving on such a trajectory. Obviously if you probed the system on time scales and length scales comparable to the characteristic scales between measurements made by the "environment" on the system then the story would be quite different. We do this kind of coarse-grain analysis all the time, not just in QM, but also in e.g. statistical mechanics; of course in QM one must be quite careful conceptually when making such coarse-grained approximations.

You yourself explained this very well in post #76 and post #94 by vanhees is also spot on. Given that you've already understood the situation at hand, what exactly do you find is at arms with what mfb is saying?

 

[TrickyDicky]

I think we are all basically on the same page on this, I was simply stressing that instead of giving preference to particles over trajectories when one is given that choice one could also consider the existence of trajectories therefore letting go of particles as ontic objects(like is done to a certain extent in QFT).

 

[mfb]

This is a very good question that probably is behind the OP in part. More specifically, is the concept itself helpful or is it hampering further developements of a more fundamental theory?

"Quantum microparticle" is not a usual expression, so how can we tell what you mean by that?

I'm not sure what you mean by "you can neglect QM" here. What is observed in particle detectors must be in principle compatible with the Standard model if one is using it to explain it.

See the comparison to gravity.
Our observations must be compatible with gravity as well, but we still don't have to care about it as its influence is negligible. The same applies to QM for tracks in a bubble chamber.

Or are you making a point about the difference between QFT and QM?

No.

 

[TrickyDicky]

"Quantum microparticle" is not a usual expression, so how can we tell what you mean with that?

Sorry, I thought the meaning was obvious by the thread's context(electrons, alpha particles), just the particles that particle physicists deal with.

 

[DavidLloydJones]

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

The exact difference is that wave functions exist in our minds, and are therefore as smooth as we feel like making them. We can dream up functions with almost no limit, and do collaborative work around any of them we have symbols to convey. Bubbles, on the other hand, go bub-bub-bub in the "real world," with a size and speed of formation crucially related to the temperature and viscosity of their medium. The individuality of the bubbles strongly reinforces, I would guess, any bias our minds have to an epistemology of distinct little round objects.

You remind me, incidentally of a fond moment now fifty years ago. Frank Oppenheimer was a friend, and my wife and I helped him and Jackie, as best we could, with the earliest stages of putting the Exploratorium together. In those fine pre-clean-and-polished-itude days, one of my favourite parts of the whole thing was a cloud chamber Frank had made out of a cardboard box, a bottle of carbon tet, and a cellophane covering to watch through.

-dlj.

 

[Bruno81]

Why is it surprising that a quantum system behaves like a particle when its position can be known in the bubble chamber? When have we seen anything to the contrary?

 

[AlexCaledin]

The quantum system's "job" seems to be Event Coordinator...

Events must be simulating particles and things...