Uncertainty in Classical trajectories

[Duda Jarek]

(...) the mention of Heisenberg and Schrödinger means you are not discussing a classical system.

I disagree - classically we often also haven't full knowledge and to cope with it we use thermodynamical models which represent our knowledge, do you disagree?

For example assuming Boltzmann distribution among possible scenarios: trajectories, what leads exactly to stationary probability distributions from Schroedinger's picture (like Feynman path integrals in imaginary time http://www.worldscibooks.com/etextbook/4443/4443_chap3_2.pdf ).

 

... and generally I would like to understand why when I've tried to discuss results claiming good agreement with experiment from dozens of peer-reviewed papers (with hundreds of citings) about such classical systems in e.g proton's EM field, you've classified it as speculations?

http://www.scienceforums.net/topic/51199-any-comments-about-gryzinski-free-fall-atomic-model/

 

 

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Moderator Note

split from charged particle in magnetic field thread

Edited September 23, 2010 by swansont

[swansont]

I disagree - classically we often also haven't full knowledge and to cope with it we use thermodynamical models which represent our knowledge, do you disagree?

For example assuming Boltzmann distribution among possible scenarios: trajectories, what leads exactly to stationary probability distributions from Schroedinger's picture (like Feynman path integrals in imaginary time http://www.worldscibooks.com/etextbook/4443/4443_chap3_2.pdf ).

 

That doesn't apply here. This is not a discussion about thermodynamics.

 

... and generally I would like to understand why when I've tried to discuss results claiming good agreement with experiment from dozens of peer-reviewed papers (with hundreds of citings) about such classical systems in e.g proton's EM field, you've classified it as speculations?

http://www.scienceforums.net/topic/51199-any-comments-about-gryzinski-free-fall-atomic-model/

 

I explained why in that thread.

 

Stop with the thread hijacking already. It's against the rules

[Duda Jarek]

I apology for misunderstanding - I'm not satisfied with the reason you gave, but I had no intention to take this discussion here - I also generally wanted by the way to understand your way of thinking in which classical trajectories are fine, unless we are considering EM field of proton, where the only description you tolerate is probabilistic(?).

My intention was to point jerryyu to Penning trap for deeper understanding of the situation he was asking for and to remind to be careful about such classical (hard) descriptions - that in practice we don't/cannot have full information, so we should rather use probabilistic (soft) descriptions like Schroedinger's picture.

 

To be more concrete - let's look at the topic: imagine we want to place a single particle on circulatory orbit perpendicular to magnetic field...

Our particle sources are not perfect - we probably can assume some Gaussian of its initial momentum and time/position - we could predict well let say a few first periods, but the initial uncertainty becomes more and more essential (so called chaos) - especially on the axis along magnetic field: we can only assume wider and wider Gaussian there - the amount of information we have is decreasing with time (so called 2nd law of thermodynamics).

 

How would you predict results of such experiment?

Do you think we can ignore our lack of knowledge while considering such classical trajectories?

peace

Edited September 13, 2010 by Duda Jarek

[swansont]

If your combined precision of the measurement of x and p exceeds Planck's constant by a few orders of magnitude, quantum effects don't matter.

[Duda Jarek]

Matter of size? Really? So how would you explain stationary probability density of electrons in macroscopic semiconductor sample? I would say that it's Schroedinger's ground state probability density ... (?)

And I would gladly finally heard which quantum effect are you referring to, to reject quantum superposition of classical trajectories picture ... or how do you understand corpuscular half of duality ... but I agree it's not for this thread ...

Returning to simple 'classical' situation from the topic - it's you who should said that it can be also described in Schroedinger's picture ... but I think such wavefunction would be thickening - it's not like in atoms that we have orbital of fixed wavefunction amplitudes, but it would be rather seen as continuously decohering ... (?)

 

It's one of reasons I didn't want to go into QM here - I agree that classical picture is more convenient here.

But still in classical picture we have unavoidable uncertainty, which through different kind of chaos leads to that for example in such macroscopic Penning trap, the most convenient picture could be thermodynamical one - it should reach some thermodynamical equilibrium and what we should work on are time averages of trajectories probability density, do you disagree?

And I would say that natural thermodynamical assumption in such 'prisoned trajectories' situation is Boltzmann distribution among possible ones - there is a funny coincidence that it leads exactly to Schroedinger's ground state stationary probability density ... so I would conclude that it's not true that Schroedinger's picture works only in Planck's scales, but is just universal - works in proton's potential, but also in macrosopic Penning trap, semiconductor ... do you disagree?

Edited September 13, 2010 by Duda Jarek

[swansont]

Matter of size? Really? So how would you explain stationary probability density of electrons in macroscopic semiconductor sample? I would say that it's Schroedinger's ground state probability density ... (?)

 

 

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Moderator Note

I don't see how that's an example of a particle in a magnetic field, so I have to wonder why your response to my request that you stop hijacking the thread is to continue to try and hijack the thread?

[Duda Jarek]

What?

I've understood your answer that the only picture in proton's potential you can tolerate is probabilistic one, because of its Planck scale size - so I've responded that it's nonsense - Schroedinger's probabilistic picture is universal - works also in macroscopic samples ... and it doesn't have to be guessed, but can be mathematically derived ...

... and yes - electron's behavior in semiconductor is also in EM field - of concrete lattice of atoms ... and as you could see - I've returned in my post to classical picture - as you've been taught for this situation ...

But I should get used to that you are not capable to discuss, but only to present your faith ... I apology for bothering you. bye

Edited September 13, 2010 by Duda Jarek

[Cap'n Refsmmat]

He'd be quite happy to discuss, if you didn't use a thread on a completely different subject to hold the discussion in.

[Duda Jarek]

So you also think that while considering (e.g. to predict something) classical trajectories, we can ignore the fact that we have limited knowledge/precision? Because for uncertainty you have QM and only QM?

It for example says that practically never we will get perfect (classical) circulation from the topic ... and that our knowledge about particle's position along magnetic field decreases with time ...

I thought that it's physicists who should be mainly worried about it ... while surprisingly it's only mathematicians who not only cares, but even treat this subject extremely seriously - we have chaos theory with Lyapunov exponents in the first approximation, then trajectories can ergodically cover some sets ... finally stabilizing density (time average) when bounded like in Penning trap...

[darkenlighten]

 

It for example says that practically never we will get perfect (classical) circulation from the topic ... and that our knowledge about particle's position along magnetic field decreases with time ...

 

It is not quantum mechanics that causes a "non-circle", but the properties of Electrodynamics.

 

Where is the probability or uncertainty in this specific subject? It's only a matter of [math] \mathbf{F} = q\mathbf{v}\times\mathbf{B} [/math] and the decaying circle in non-ideal situation is due to the radiation decay, as stated before.

[Duda Jarek]

No - I'm saying that classically we also don't have infinite precision - we know parameters with some unavoidable uncertainty - and it often grows with time (along directions with positive Lyapunov exponents) ... for example even in idealized situation from the topic, we still have decrease of knowledge (with time) about the current state of system ...

Or in other words - butterfly effect doesn't need quantum mechanics ...

Radiation decay is succeeding complication - we can rather only predict probability of such phenomenas - so while considering statistical ensemble among all possible scenarios for our knowledge of initial conditions, we should add to this set scenarios with decay using proper probability density ...

Classical mechanics may look simpler than quantum ... but it's misleading - to do it seriously, we just have to consider that we don't have full information and so use some probabilistic/thermodynamical models - making calculations quite similar to quantum ...

[swansont]

Or in other words - butterfly effect doesn't need quantum mechanics ...

 

 

No, it doesn't, but what it does need is a nonlinear system. If you want to make the case that the Lorentz force qualifies, go ahead and try.

[Duda Jarek]

Lagrangian mechanics conserve volume, so indeed practically any nonlinearity have also repelling direction, making that we lose information exponentially with time.

But is it really necessary for information loss? For example linear one?

Let's imagine the simplest situation: you have a single particle in empty space (no interactions) and you know its initial velocity with finite precision - how does you knowledge about its position evolve with time? (or your knowledge about position along magnetic field from the topic...)

 

ps. And in practice we should also remember that e.g. electron have magnetic momentum, what complicates the dynamic through precessive motion - adding nonlinearities ...

ps2. I see that since QM, even if it imply classical mechanics (Ehrenfest theorem), physicists are no longer interested in it, so I would like to advertise one of the best books I've studied and the only good one in classical mechanics I've seen - giving not only standard for physicist tool to shut up and calculate, but really deep understanding, showing its beauty - Arnold's books.

Edited September 15, 2010 by Duda Jarek

[swansont]

 

Let's imagine the simplest situation: you have a single particle in empty space (no interactions) and you know its initial velocity with finite precision - how does you knowledge about its position evolve with time? (or your knowledge about position along magnetic field from the topic...)

 

 

 

Classically, "knowledge about the position" and "position" do not mean the same thing. Not knowing the exact energy of the particle does not mean that it won't travel in a circular path in a magnetic field.

[Duda Jarek]

But Heisenberg's quantum uncertainty also doesn't mean that there is no internal dynamics - it only restricts practical aspect: measurement.

Taking further implications based on Bell's inequalities assumes that probabilistic models on local deterministic Lagrangian (field) mechanics should be also local - such assumption about models representing our knowledge is just wrong (see maximal entropy random walk).

 

In not standard: idealistic, but practical classical mechanics we also have to have in mind that we cannot have infinite measurement precision, full information - we have to work on probability clouds ... what through chaos, ergodicity leads to practical picture far from the idealistic one ...

[Duda Jarek]

Basically this classical-quantum correspondence we were talking about is the region of so called quantum chaos - I believe MERW-based model is what they were missing to understand it - in this presentation are for example its 2 intuitive derivations and connection with quantum chaos.

 

If someone is interested, here has developed discussion about MERW: http://www.sciforums.com/forumdisplay.php?f=33

Edited December 7, 2010 by Duda Jarek