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From what I understand, the uncertainty principle states that there is a fundamental natural limit to how accurately we can measure velocity and momentum at the same time. It's not a limit on equipment but just a natural phenomenon.

However, isn't this just an observational limit? There is a definite velocity and momentum, we just don't know it. As in, we can only know so much about the universe, but the universe still has definite characteristics.

Considering this, how do a wide range of quantum mechanical phenomena work? For example, quantum tunneling - its based on the fact that the position of the object is indefinite. But the position is definite, we just don't know it definitely. Or the famous light slot experiment? The creation of more light slots due to uncertainty of the photon's positions?

What I am basically asking is why is a limit on the observer, affecting the phenomenon he is observing? Isn't that equivalent to saying because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation!

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+1 a nice question.. –  Vineet Menon Apr 20 '12 at 6:02
    
This is an almost-duplicate of : physics.stackexchange.com/q/19500 . I tend to be charitable in identifying duplications, but my answer there applies here. –  Ron Maimon Apr 20 '12 at 8:06
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@RonMaimon I think that the almost is enough differentiation. –  user8614 Apr 20 '12 at 12:23
    
Uncertainty can be described in terms of commutation relations. This may not be an answer you're looking for, but it's a pointer that it exists in the maths if you start looking for it. –  P O'Conbhui Apr 21 '12 at 4:12
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There is a definine velocity and momentum, we just don't know it.

Nope. There is no definite velocity--this was the older interpretation. The particle has all (possible) velocities at once;it is in a wavefunction, a superposition of all of these states. This can actually be verified by stuff like the double-slit experiment with one photon--we cannot explain single-photon-fringes unless we accept the fact that the photon is in "both slits at once".

So, it's not a knowledge limit. The particle really has no definite position/whatever.

Isn't that equivalent to saying because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation!

No, it's equivalent to saying "because we haven't gotten any evidence of Star X, it may or may not exist --it's existence is not definite" Technically, an undetected object does exist as a wavefunction. Though it gets slightly philosophical and boils down to "If a tree falls in a forest and no one is around to hear it, does it make a sound?"

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You say "The particle really has no definite position/whatever", and then you say "Star X ... may or may not exist". Your attitude is inconsistent. –  Mitchell Porter Apr 20 '12 at 22:36
    
@MitchellPorter: I've mentioned that it does exist as a wavefunction, but I clarified it anyway. –  Manishearth Apr 21 '12 at 1:29
    
If a tree falls in a forest that may or may not exist, does it make a wavefunction? –  P O'Conbhui Apr 21 '12 at 4:08
    
@PO'Conbhui the forest is a wavefunction, in which it may or may not exist. The inhabitants of this psuedoforest are part of the wavefunction. The event of a tree falling is also part of the wavefunction. –  Manishearth Apr 21 '12 at 4:30
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Manishearth's answer is correct, and this is just a minor extension of it. Manishearth correctly points out that the problem is your statement:

There is a definine velocity and momentum, we just don't know it.

Your statement is the hidden variables idea, and courtesy of Bell's theorem we currently believe that hidden variables are impossible.

Take the example of a hydrogen atom, and ask what the position of the electron is. The problem is that properties like position are properties of particles. It doesn't make sense to ask what the position is unless there is a particle at that position. But the electron is not a particle. The question of what an electron really is may entertain philosophers, but for our purposes it's an excitation in a quantum field and as such doesn't have a position. If you interact with the electron, e.g. by firing another particle at it, you will find that the interaction between the particle and electron happens at a well defined position. We tend to think of that as the position of the electron, but really it isn't: it's the position of the interaction.

The uncertainty principle applies because it's not possible for an interaction, like our example of a colliding particle, to simultaneous measure the position and momentum exactly. So you're sort of correct when you say it's an observational limit, but it's a fundamental one.

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Knew there was a word for the interpretation! Just couldn't remember. :) –  Manishearth Apr 20 '12 at 6:34
    
I always like to use the qualifier local for Bell's theorem. For the 1.5 people of us who like to believe somewhat in the Bohm interpretation, it does carry some weight. </pedantic> –  Muhammad Alkarouri Jun 8 '12 at 17:54
    
We tend to think of that as the position of the electron, but really it isn't: it's the position of the interaction. Many textbooks say when you measure the state of a particle, it's really in that state at the moment of observation. Does it really make a difference? –  jinawee Apr 2 at 13:27
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This seeming leap is an invocation of logical positivism. Logical positivism is the default philosophy in physics, it is indispensible, and it has been the source of nontrivial ideas which have been crucial to progress for over a century.

You can't assume that there is a position and momentum simultaneously in the particle, because this point of view would lead you to believe that there is a probability for the position and the momentum, and that each possible position and momentum evolves independently. This is incompatible with observations. No independent position, momentum picture can be different from Newtonian, classical mechanics.

You can see this, because a wavepacket with a nearly definite momentum moves as the classical particle, a wavepacket with a nearly definite position is at a spot, like a classical particle, so together, if both are well defined at all times, the particle would be moving from definite position to definite position as in classical mechanics. This is impossible, because it would lead to sharp trajectories and no diffraction of electrons around objects. Electron diffraction is observed every day.

One can still claim that the position is a hidden variable, and not the momentum, but then the momentum is only partly defined, as a property of the carrier wave. This is what happens in the Bohm theory.

The reason one cannot assign hidden variables in an obviou way to the particles in quantum mechanics is because the calculus regarding the different possibilities is not a probability calculus, but a calculus of probability amplitudes, and probability amplitudes don't have an ignorance interpretation.

To see this, consider a particle which can goes from a state $|0\rangle$ (where some physical bit describing its position is 0) to the superposition state $|0\rangle+|1\rangle$ and from $|1\rangle$ to $-|0\rangle + |1\rangle$ in a certain period of time, say 1 second. Now start off in the state ($|0\rangle+|1\rangle$), what happens? By linearity, you end up in the definite state $|1\rangle$. So if you consider "1" a definite state, it becomes more uncertain, but in the uncertain combination, it recongeals to become certain! This doesn't happen in probability, because different probabilistic branches can't combine with a minus sign the way that they just did above in quantum mechanics to get rid of the $|0\rangle$ component.

The sign issue makes the ignorance interpretation of quantum mechanics untenable--- only probabilities are ignorance, and only in the limit of very large systems does quantum mechanics (approximately) reproduce something like probability. This involves observation.

The way in which the theory was constructed was by carefully applying logical positivism at each stage, and if one does not internalize and apply positivism, you don't get the theory. See this related answer: How can indeterminacy in quantum mechanics be derived from lack of ability to observe a cause?

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Quantum mechanics describes everything with "wavefunctions" or "state vectors" that provide probabilities for position, velocity, etc. As Caraiani Claudiu says, the mathematical details make it impossible for a wavefunction to provide a probability of 100% for a particular position and a particular velocity.

Also, it has been proved in many ways that it is very difficult to make a deeper theory that explains quantum mechanics - to propose a new set of laws which give rise to an average behavior that matches quantum mechanics. The theory of David Bohm is the simplest approach to this, but it is hard to extend it to relativity and to fermions.

So most physicists attempt to believe that quantum mechanics is the final framework of physics, and they construct rationalizations for this intellectual position. They will say that nothing is real until you measure it, or that the electron does everything at once until you look at it (and then you see it doing just one thing), or that by definition we don't see what we don't see so we shouldn't care if the theory offers no coherent account of what happens between measurements.

Among the attempts to make sense of quantum mechanics, I should also reserve a special mention for the belief in "many worlds", according to which all the possibilities in the wavefunction are equally real and happening in separate parts of a "multiverse". At least this looks like an attempt to restore an objective conception of reality, without verbal games. However, if you examine the details, you will find that there is no "many worlds theory" in the sense of a coherent, self-sufficient set of concepts. Possibly there could be a many-worlds theory one day, but at the moment it is just another wall of words.

There are several reasons for the persistence of this pathological situation.

First, quantum mechanics works very well. It doesn't just make successful predictions, it is a framework which can be extended to include new particles and new types of interaction, without abandoning the uncertainty principle and all the other features which make it unsatisfactory as an ultimate theory.

Second, although it does not offer a conceptually coherent account of objective reality, it does offer a coherent self-contained framework for making predictions about observable phenomena, if that is all you are interested in.

Third, the mathematical difficulty of fundamental physics is such that people have no room in their heads for also trying to explain quantum mechanics itself. The people trying to do that, with very few exceptions, are usually not working on the most advanced theories.

And fourth, it must just be difficult to discover the truth about this. We may have to develop some entirely new set of concepts by means of which to understand basic entities and their properties. There may be no particles, and there may be no position or velocity as presently imagined. Those may simply be informal common-sense concepts, pushed into domains where they don't actually apply. Whenever you hear someone saying, quantum mechanics definitely implies some particular picture of objective reality (or even worse, saying that it implies that there is no objective reality), you aren't hearing truth, you're just hearing dogma, the desire of a human being to be in possession of the truth even when they aren't.

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People tend to say,this is Heisenberg principle,that this something deep,but people tend to forget something,this is a math fact and it was way before Heisenberg in classical fourier analysis.It's like saying k sin^2(x) + k cos^2(x)=k it is your principle. Well,remove the k and you`ll that this principle was way long before you.

http://www.nato-us.org/analysis2000/papers/havin.pdf

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You are confusing the mathematical Heisenberg principle with the physical one. They are related mathematically, but the physics is important in establishing the physical validity of quantum mechanics. And Fourier transforms were relatively new in Heisenberg's time, and the product relation of widths was not known before 1925 as far as I know. –  Ron Maimon Apr 20 '12 at 15:24
    
And what exactly am I confusing?:) –  Caraiani Claudiu Apr 20 '12 at 16:01
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The statement: "the variance of a distribution is bounded below by the variance of the Fourier transform" which is the signal-processing uncertainty principle, mathematically, and the physical statement that the state of a particle cannot have a certain position and momentum, which logically requires you to identify momentum and position as having probability amplitude distributions which are Fourier transforms of each other. The two ideas are related, but the mathematics does not obviously imply the physics, because the physics is physical, it requires a link to reality. –  Ron Maimon Apr 20 '12 at 16:51
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