Uncertainty in calculations

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ansgar
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Uncertainty in calculations

Post by ansgar » Fri Dec 18, 2009 9:49 am

It might seem like an odd and rather naive question, but I still have
to ask it.

When describing motion of particles at the LHC do researchers still use calculus or related methods? And is it still possible to accurately calculate the momentum and position of a particle. Even at that level, or even at smaller levels, say even below Planck length.
And I mean "calculate it accurately", and not "measure it accurately". I am aware that there are limits on measuring momentum and position. But are there theoretical limits on calculating the position or momentum. I'd be very thankful for a reply.

Cheers

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Anitusar
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Re: Uncertainty in calculations

Post by Anitusar » Fri Dec 18, 2009 10:40 am

Which particles do you mean?

The protons in the beam are not treated individually. It's also not neccesary.

And all reconstructed particles in the experiment are using measurements. You can calculate back wards using the measured paricles to get the particles which decayed (reconstruct their mass and momentum), but all calculation are then based measurements with all its errors.

Generally speaking, the researches are treating this whole as statistical samples and are not so much interested in the individual particle.

Also, getting to this low scales, its harder to define position of a particle. In Quantum mechanics, this gets turned into probability density (i hope this is the right english term). One can only say that the particle is at a position x with a certain probability (that is if you do some kind of measurement, otherwise it is not really defined).

For example, take the Hydrogen atom (one proton, one electron):
You can not calculate a path for the electron on that it flies around the proton. You can just calculate the probabilty to find the electron at a certain position when you do a measurement. Also if you do a second measurement, one can not calculate back the path the electron had taken to get from a to b.

ansgar
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Re: Uncertainty in calculations

Post by ansgar » Fri Dec 18, 2009 11:26 am

Thanks for the answer, it brought up many interesting aspects. But an important aspect of of the question is still open, maybe because it is an uncommon question. The question is whether the mathematics is still correct when applied to the sub-quantum level. Regardless of the particle that was described. Can you apply e.g. calculus even for phenomena at the smallest level? Or do the computed values become incorrect?

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DCWhitworth
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Re: Uncertainty in calculations

Post by DCWhitworth » Fri Dec 18, 2009 12:23 pm

I'm curious, why do you think calculus might not work at the sub-atomic level ? It's a mathematical process not a theory or law to understand the physical world.
DC

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ansgar
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Re: Uncertainty in calculations

Post by ansgar » Fri Dec 18, 2009 12:35 pm

The argument would be that infinite-series solutions in e.g. calculus, are based on strict, hard 1:1 correspondences of physical locations with physical objects. Wouldn't this be contrary to quantum mechanics?

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Re: Uncertainty in calculations

Post by josch222 » Fri Dec 18, 2009 5:31 pm

ansgar wrote:The argument would be that infinite-series solutions in e.g. calculus, are based on strict, hard 1:1 correspondences of physical locations with physical objects. Wouldn't this be contrary to quantum mechanics?
Of course you need different math. formulas to describe different things,
or more apt for this examples: For different cases of how exactly
you have to describe something accurate enough for the given purpose.
Its the same witch classical, newtonian mechanics and relativistic mechanics.
If you go from the "near the speed of light" case to slower movements,
the relativistic terms becomes more and more irrelevant.
So you can leave them out and you end up with a special case formula for daily
life (I mean the daily life of a non physicist ;-).
If you go from the quantum scales and the more generally formulas to bigger
objects you will end up with the same simple formulas for common scales.

But math as itself remains the same, there is only one math, derived from logic
(and a few axioms). It works great for nearly everything and nobody came up
with a "new math" so far, there were only extensions. I would go so far to say that
anything not consistent with the known math must be illogical.

Joerg.

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Re: Uncertainty in calculations

Post by Xymox » Sat Dec 19, 2009 9:26 am

:) See Ansgar ! I told you your question would be best answered in the forum !

ansgar
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Re: Uncertainty in calculations

Post by ansgar » Sat Dec 19, 2009 9:35 am

So, you to make sure it is understood correctly.

Imprecise measurements of parameters or initial conditions, might lead to imprecision of a prediction. And it might be possible that a model is incomplete, like not taking into account relativistic effects, which may lead to wrong outcomes. And also concepts such as "position" may not apply to all systems, such as quantum system. Is this correct?

But it would be wrong to claim that you cannot calculate the position of an object, regardless of scale, because of quantum mechanics. Is this claim just somewhat wrong, or fundamentally wrong? Or am I reading you replies wrongly and is it correct that you cannot calculate the position?

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Re: Uncertainty in calculations

Post by Anitusar » Sat Dec 19, 2009 11:05 am

ansgar wrote:So, you to make sure it is understood correctly.

Imprecise measurements of parameters or initial conditions, might lead to imprecision of a prediction.
Yes. There is even a uncertainty principle. It says, that you cannot measure two correlated thinks with infinite precision. For example, if you want to measure the position of a particle with great precision, you cannot measure its momentum at the same time with that precision.
ansgar wrote:And it might be possible that a model is incomplete, like not taking into account relativistic effects, which may lead to wrong outcomes.
Exactly. That's why you try to create "extrem" situations in order to see if your model still works.
ansgar wrote: And also concepts such as "position" may not apply to all systems, such as quantum system. Is this correct?

But it would be wrong to claim that you cannot calculate the position of an object, regardless of scale, because of quantum mechanics. Is this claim just somewhat wrong, or fundamentally wrong? Or am I reading you replies wrongly and is it correct that you cannot calculate the position?
You can calculate it, the answer just will not be a single (infinite small) point in space.


Edit: Maybe one more remark. The problem at such small scales is, that you somewhat lose the concept of a particle as a solid hard object. On these scales, the electron (and every other particle) acquire the properties of a wave. This makes the concept of a single infinite small point as the position of the particle no longer applying to this situation.

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Re: Uncertainty in calculations

Post by ansgar » Sat Dec 19, 2009 11:36 am

Anitusar wrote: You can calculate it, the answer just will not be a single (infinite small) point in space.
So, the answer will be a distribution, but it is not the case that that the calculus is in itself incorrect?
Anitusar wrote: Edit: Maybe one more remark. The problem at such small scales is, that you somewhat lose the concept of a particle as a solid hard object. On these scales, the electron (and every other particle) acquire the properties of a wave. This makes the concept of a single infinite small point as the position of the particle no longer applying to this situation.
[/quote]
So, it doesn't make sense to ask what the exact position of a wave is?

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Re: Uncertainty in calculations

Post by Anitusar » Sat Dec 19, 2009 12:15 pm

ansgar wrote:
Anitusar wrote: You can calculate it, the answer just will not be a single (infinite small) point in space.
So, the answer will be a distribution, but it is not the case that that the calculus is in itself incorrect?
Anitusar wrote: Edit: Maybe one more remark. The problem at such small scales is, that you somewhat lose the concept of a particle as a solid hard object. On these scales, the electron (and every other particle) acquire the properties of a wave. This makes the concept of a single infinite small point as the position of the particle no longer applying to this situation.
ansgar wrote:So, it doesn't make sense to ask what the exact position of a wave is?
The position of the particle is described by a wave function. Depending on the question you want to answer, you have to treat the electron as a solid object or as a wave. The double slit experiment is one example for that.

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Re: Uncertainty in calculations

Post by Xymox » Sat Dec 19, 2009 1:32 pm

ahhh... the infamous double slit...

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Re: Uncertainty in calculations

Post by tswsl1989 » Sat Dec 26, 2009 8:19 pm

ansgar wrote:
Anitusar wrote: So, the answer will be a distribution, but it is not the case that that the calculus is in itself incorrect?
I'd say no, the calculus is not incorrect
Calculus is pretty essential in particle physics and quantum mechanics, at least at the level I'm studying it atm.
ansgar wrote:
Anitusar wrote: Edit: Maybe one more remark. The problem at such small scales is, that you somewhat lose the concept of a particle as a solid hard object. On these scales, the electron (and every other particle) acquire the properties of a wave. This makes the concept of a single infinite small point as the position of the particle no longer applying to this situation.
So, it doesn't make sense to ask what the exact position of a wave is?[/quote]
It makes more sense to ask about the properties of the wave at a point. Taking the double slit example, there are areas where there the resultant wave has 0 amplitude. The wave is still there, but it has no amplitude. In terms of particles, there are no particles hitting that area.

That make sense, or have I made things more confusing?

Edit: Nested quote fail

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