Why probilities and not definates.

[Wizardsblade]

I’m in intro quantum this semester so I probably just know enough to be dangerous. From what I understand a particle will have a probability of being in a certain state, we are then able check and see where it was. This puts it in another state with a certain probability.
From my understanding of probability (not necessarily QM), a probability is only a probability because not all the function is known. I.e. if I tell you a computer program will display head half the time and tails half the time. You will not be able to tell me if it will show heads or tails will be displayed, but if I told you that when the computers clock is an even hour heads will be displayed and on odd hours tails will be displayed; then you will then be able to tell me for certain if the program will display heads or tails. This seems to tell me the reason for probability is that the entire equation is not known.
So is the reason of probability in QM because the probability function is not completely know or what is the reason?

Thanks

 

[StatMechGuy]

The reason is because quantum mechanics is a probabilistic theory. Asking "why is it so?" is like asking why Newton's laws are what they are. Because that's how nature decided to do things.

Truly, it's a random, statistical theory in the purest sense.

 

[christianjb]

Not a complete answer, but

The particle's wavefunction is not in general in any eigenstate. It's only true that a measurement has a probability p_i of returning a result of eigenvalue e_i, where sum_i p_i = 1 and e_i is an eigenvalue of the measurement operator A.

These probabilities are not generally regarded as representing a lack of information about the system, because the system really doesn't possesses a value (position, momentum, etc.) to begin with.

 

[Anonym]

From what I understand a particle will have a probability of being in a certain state, we are then able check and see where it was. This puts it in another state with a certain probability

You almost answered your question. If a particle is in the pure state, we are then able check and see where it was. This leaves it in the same state with a certainty.

If I tell you a computer program will display head half the time and tails half the time. You will not be able to tell me if it will show heads or tails will be displayed, but if I told you that when the computers clock is an even hour heads will be displayed and on odd hours tails will be displayed; then you will then be able to tell me for certain if the program will display heads or tails.

It is too complicated description. The things are much simpler. The measurement devices are macroscopic systems and therefore statistical in nature. The single outcome has no meaning. You should collect statistical ensemble of individual observations to define the observed value(s). If a particle is in the pure state of the dynamical variable that you measure, you will get the “certain” result and even you may continue your measurements of the same particle since it remain in the same state. If a particle is not in a pure state then your measurement lead to collapse and your further measurements are not legal (you measure the same particle but in the different state). Therefore you should repeat your measurements (prepare the identical system and do measurement again and again till you get certain picture; in practice about 100000 repetitions turn out to be enough). Since your device is intrinsically statistical system you sometimes will get the head, sometimes the tail, sometime the same point as in case of CM system. After a while you will have pretty accurate picture which display the extended (“blurred”) object. This is the difference between CM and QM of the single material object (in QM this is a field and is not described by the single point in space in general). In case of the single material object (electron, photon, etc) it has nothing to do with statistics but when you consider many particle QM system it is obvious that you should use the statistical consideration.

This seems to tell me the reason for probability is that the entire equation is not known.

Here I did not understand what the entire equation you are looking for. If you mean the equation that describes the measurement process, it occurs instantly and therefore no dynamical equation is required. If you mean the equation that describes “how nature decided to do things”, ask StatMechGuy-he know.

Regards, Dany.

 

[christianjb]

Dany- I find that confusing.
You can certainly have 'normal' statistical probabilities for an ensemble of 1 particle systems, but you can no longer describe the distribution by a single wave-function. Instead you have to use the density matrix description, which naturally incorporates 'lack-of-knowledge' probabilities as discussed by the original poster.

 

[christianjb]

It's also worth remarking that even complete knowledge of the entire wavefunction can not give you certainties for results from measurement, unless in the trivial case your system is in an eigenvalue corresponding to the measurement that's about to happen.

A complete description of the wavefunction only gives you the probability i that the measurement will return a vaule ei.

 

[Anonym]

From what I understand a particle ….

:You can certainly have 'normal' statistical probabilities for an ensemble of 1 particle systems.

Correct, but Wizardsblade original question concerned a single particle QM system. I discuss above the single particle system only, I mentioned that an ensemble of one particle systems should be described statistically.

It's also worth remarking that even complete knowledge of the entire wavefunction can not give you certainties for results from measurement, unless in the trivial case your system is in an eigenvalue corresponding to the measurement that's about to happen.

A complete description of the wavefunction only gives you the probability i that the measurement will return a vaule ei.

All repeated measurements are equally legal. You use the circular arguments. Try to understand mine. Each one of your measurements is completely certain since your measurement devices belong to the Classical World. It is the collapse of wave packet all about.

Regards, Dany.

 

[christianjb]

Correct, but Wizardsblade original question concerned a single particle QM system. I discuss above the single particle system only, I mentioned that an ensemble of one particle systems should be described statistically.



All repeated measurements are equally legal. You use the circular arguments. Try to understand mine. Each one of your measurements is completely certain since your measurement devices belong to the Classical World. It is the collapse of wave packet all about.

Regards, Dany.

Uh...
I'm allowed to find your post confusing aren't I?
I don't know what a 'legal' measurement is. I don't know what you mean by a measurement device belongs to the classical world. I don't know what you mean by a certain measurement. I don't know where you think my argument is circular.

 

[Anonym]

I'm allowed to find your post confusing aren't I?

Sure.

I don't know what a 'legal' measurement is. I don't know what you mean by a measurement device belongs to the classical world. I don't know what you mean by a certain measurement. I don't know where you think my argument is circular.

So what? It is consistent.

I’m in intro quantum this semester so I probably just know enough”

My intention was to emphasize that the unitary evolution of the quantum system is deterministic and the classical physics are deterministic. Therefore time available for the assumed statistical behavior is during the measurement (delta t =0). It perhaps not a right time for him to go to W@Z and all that, therefore I did not mentioned it. If you are interesting, read there H.D. Zeh analysis and M. Born original paper (1926).

Regards, Dany.

 

[Thrice]

I was under the impression that the probability aspect is one interpretation out of many reasonable ones. I'm on my 2nd semester of QM & we've not really done anything on interpretations.

 

[FunkyDwarf]

Correct its the Copenhagen interpretation, the most widely used one, but there are others.

 

[Wizardsblade]

StatMechGuy,

I understand what you are saying, but are not scientest looking for gravatons to explain why/how gravity works? I guess what boths me is that in math a probability only comes about when a function is not completely defined (ie i tell you there is a point at R=3 theta=0 and phi=x you can tell me the probility of the point being between some phi' and phi''. If i say phi=0 then you can no longer need a probability but can just state your point is at (3,0,0). Looking at math like this it seems to say that there is some part of Schrodinger wave equation that is not fully defined. (meaning if it where defined we would remove probability from QM.) I would agree that fulling defining Schrodinger's wave equation may be imposible, but i do not think we should say QM is complete without at least trying.

Maybe I am just to CM oreinted dunno. I love the way QM and freewill play well together, I find that facinating.

 

[jtbell]

We really don't know what lies "underneath" the wave function $\Psi$ and the probability distributions that we calculate from it, or even if anything does lie underneath it. (Actually, we do know that the QM you learn as an undergraduate is an approximation to relativistic quantum field theory, but QFT still makes its predictions in terms of probabilities so it doesn't answer this question.)

As FunkyDwarf noted, there are various interpretations of QM that attempt to describe what is "really happening," but so far, they make the same predictions for the results of experiments so there's no way to distinguish between them. They all have aspects that make different groups of people uncomfortable, and there is no consensus on which one is "really correct."

If you look back through old threads in this forum, you'll find that the longest ones are arguments/discussions about the merits of various interpretations of QM.

 

[Wizardsblade]

cool, thanks jtbell

 

[tpm]

The question is..does a particle have 'linear momentum' or 'angular momentum' well defined although we can't measure it??..perhaps the particle has a well defined position and momentum but as it happens with Browninan motion due to colisions with virtual particles (??), or Fields in the vacuum then the trajectories becomes 'chaotic' and we can't measure them just as it happens with Brownian motion.

 

[christianjb]

The question is..does a particle have 'linear momentum' or 'angular momentum' well defined although we can't measure it??..perhaps the particle has a well defined position and momentum but as it happens with Browninan motion due to colisions with virtual particles (??), or Fields in the vacuum then the trajectories becomes 'chaotic' and we can't measure them just as it happens with Brownian motion.

The 'Kochen Specker paradox' demonstrates that it's impossible for the wave-function to have (definite but unknown) values for (non-commuting) observables before measurement.

 

[Anonym]

I was under the impression that the probability aspect is one interpretation out of many reasonable ones. I'm on my 2nd semester of QM & we've not really done anything on interpretations.

Correct its the Copenhagen interpretation, the most widely used one, but there are others.

Recently, I struck upon the first hand presentation of the Copenhagen interpretation which I consider remarkably clear, deep and objective:
W. Heisenberg “The Physical Principles of the Quantum Theory”, Dover Publications, 1930 (the lectures given at the University of Chicago).

Regards, Dany.

 

[muppet]

StatMechGuy,
I guess what boths me is that in math a probability only comes about when a function is not completely defined ... Looking at math like this it seems to say that there is some part of Schrodinger wave equation that is not fully defined. (meaning if it where defined we would remove probability from QM.) I would agree that fulling defining Schrodinger's wave equation may be imposible, but i do not think we should say QM is complete without at least trying.

I don't think that's technically true. In my course in probability, an uncertainty inherent in the system was taken as axiomatic to everything we did. I take your point that in mechanics if you specify the initial conditions of a coin toss (say) very precisely then you can predict the outcome with 100% accuracy; but if you take uncertainty as "built-in" then a probability distribution or density function can still be "fully defined", it just can't be used to predict outcomes with deterministic accuracy.

It's perhaps worthwhile to note that Schroedinger didn't construct his wave equation as a probabilistic tool; he derived it as a deterministic equation that governs the evolution in time of the wave associated with the particle which then as now was a nebulous idea; it was Born who said that the wavefunction was linked to a probability density function. I think a lot of professional physicists genuinely believe that the Copenhagen interpretation is actually representative of the ultimate level of reality; the time those who object to it spend trying to find a better description, they spend wondering how the wavefunction collapses

 

[Demystifier]

The 'Kochen Specker paradox' demonstrates that it's impossible for the wave-function to have (definite but unknown) values for (non-commuting) observables before measurement.

He asked about the particle, while your answer refers to the wave function. It is possible that the particle is not the same thing as the wave function. It is also possible that the particle has some values of non-commuting variables even before measurement, but that these values change by the act of measurement, implying that the measured values are not equal to those before measurement.

 

[HallsofIvy]

From my understanding of probability (not necessarily QM), a probability is only a probability because not all the function is known. I.e. if I tell you a computer program will display head half the time and tails half the time. You will not be able to tell me if it will show heads or tails will be displayed, but if I told you that when the computers clock is an even hour heads will be displayed and on odd hours tails will be displayed; then you will then be able to tell me for certain if the program will display heads or tails. This seems to tell me the reason for probability is that the entire equation is not known.

It's your understanding of probability that is incorrect. The example you have given is NOT random, it is deterministic. The question of whether the exists such a thing as a purely random event that is NOT deterministic at some "deeper" level is one of the basic questions of quantum theory, not probability which says nothing on that subject.