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I ask this question because I have read two different quotes on the uncertainty principle that don't seem to match very well. There are similar questions around here but I would like an explanation that reconciles these two interpretations specifically:

  1. Feynman talks about the uncertainty principle in one of his lectures and mentions it as the reason why electrons don't crash into the atom's nucleus: If they did they would have an exact location and momentum which is not allowed by the uncertainty principle. In saying this it is clear that the uncertainty principle is a fundamental property of nature because it has an effect on where an electron can reside.

  2. Recently I read - somewhere else but I forgot where exactly - an account of the uncertainty principle where there was explained how we can measure position of a particle by firing another particle into it, the collision disturbs the velocity of the observed particle therefore we can not know its momentum anymore.

Now, 2) very much seems like a limitation of what the observer can know, while 1) attributes a fundamental property of nature to it (electrons don't crash into the nucleus). What is the correct way to think about this?

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Strictly speaking there is currently no unique answer on this question even between main stream physicists! and many of them puts this question in "Metaphysics" category because, essentially, any of those two views will not affect the theoretical prediction that actually reflects reality to a very high accuracy. –  TMS Feb 17 '13 at 10:59
    
Ok I see, that helps thanks! And in the case it isn't a property of nature we don't know why an electron doesn't crash into the nucleus, correct? –  Jeroen Moons Feb 17 '13 at 16:28
    
Possible duplicates: physics.stackexchange.com/q/24068/2451 and links therein. –  Qmechanic Feb 17 '13 at 19:06
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2 Answers 2

up vote 8 down vote accepted

The short answer is : it is a fundamental property of nature.

The very short answer is "quantum"

The long answer:

From the beginning of the 20th century, slowly but certainly Nature revealed to us that when we go the very small dimensions its form is quantum. It started in the middle of the nineteenth century , with the table of elements which showed regularities that could not be explained except by an atomic model with equal electrons to the charge of the nucleus.

There were efforts to understand why the electrons which were part of the atoms did not spiral down into the nucleus and disappear, with the Bohr model . This introduced the idea of "quantum". The energy the electrons were allowed to have in the possible orbits around the nucleus was postulated to be quantized. In a similar way that the vibrations on a string have specific frequencies allowed with wavelengths which are multiples of the length of the string, the electrons about the atoms could have only specific energies.

Then a plethora of experimental results led theorists to postulate quantum mechanics from a few "axioms" . Starting with the Schrodinger equation formal theoretical quantum mechanics took off and we never looked back because it fits perfectly all known experimental data in the microcosm, and not only.

The uncertainty principle is a lynch pin in the mathematical formulation of quantum mechanics.

A premiss is that all predictions of the QM theory are given as probability distributions, i.e. no observable can be predicted except as a probable value.

In quantum mechanics to every physical observable there corresponds an operator which acts on the state functions under study. Operators often are represented by differential forms and the algebra of operators holds. In quantum mechanics two operators can be commuting, that is they can be like real numbers a*b-b*a=0, or not, the value can be different than 0. This means that one is working in a larger set than the real numbers, complex numbers are needed.

The Heisenberg uncertainty principle for position and momentum as it appears in the fundamental postulates of quantum mechanics is a commutation relationship between conjugate variables, x and p, represented by their corresponding operators:

$$[x,p_x]=i\hbar$$

This relationship is very fundamental in the theory of Quantum Mechanics which describes very successfully matter as we have studied up to now, mathematically. If the HUP were falsified it would falsify the foundations of QM.


Now on the subject of the electron and the nucleus. The quantum mechanical solutions that describe the orbitals of the hydrogen atom, for example, have non zero probabilities for the electron to find itself in the center of the nucleus, when the angular momentum is zero. So it is not clear to me how Feynman could have used that hand waving argument you are describing in your question. After all we do have electron capture nuclear reactions. He is probably basing the argument on the very small volume the nucleus occupies with respect to the atomic orbitals which will give a very small probability of capture.

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Thank you very much, that is very illuminating! I'm doing self-study from the Feynman Lectures, I still have a long way to go :) –  Jeroen Moons Feb 17 '13 at 16:31
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-1: What you wrote is the canonical commutation relation for position and momentum; the uncertainty relation is $\sigma_x \sigma_p \geq \hbar/2$. –  joshphysics Feb 17 '13 at 19:55
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@joshphysics I was already talking of operators and in operator language that is how the Heisenberg uncertainty principle comes out and is basic in the QM formulation. stanford.edu/~rsasaki/AP387/chap1 see section 1.2.2 where it is derived from the commutation relations. –  anna v Feb 17 '13 at 20:37
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@annav Yeah I mean I understand that the operator algebra is used to derive the generalized uncertainty relation, and I think you make a valid point, but I think that as a semantic issue, it's a bad idea to refer to cummutation relations as uncertainty relations because it will confuse people who aren't familiar with this fact. –  joshphysics Feb 17 '13 at 20:53
    
@joshphysics I will try to clarify the connection. It is important on how the HP is fundamental: because conjugate variables are fundamental postulates in QM. If HP goes QM has to be modified –  anna v Feb 18 '13 at 5:28
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This is an interesting question. It deserves an answer without mathematical complications!

Uncertainty is not an anthropocentric phenomenon

Laws of Nature

To get some understanding of this one must understand one thing: Whatever happens in nature, whether in the animate or inanimate world (including ourselves), there are rules that dictate how things will happen, we call them the laws of nature, and these are what scientists are trying to discover and understand in as much depth as possible. We only discover approximately what these are, however, by building models through which we are trying to get as close to reality as we can or are “allowed to” by our limited ability of observation and brain capacity.

Experiment

When we do an experiment, we are in a constant ‘dialogue’ with nature, and sometimes we even ‘provoke’ her to see how see will respond so that we can get closer to her secrets! The models of physics which we develop in order to explain the outcomes of our ‘dialogue’ with nature, will inevitably contain numbers (the physical constants) which help us put some order in our conversation with nature. Planck’s constant is one of those physical constants. Without it, nothing we have learnt during our conversation with nature would make sense! The evidence that such physical constant is real, does exist and makes sense, comes from the continued conversation we are having with nature (the outcomes of our experiments), she ‘allows’ us to measure it and shows us almost every corner of the world where she is using it. The uncertainty principle is yet another rule of nature, which she imposes through Planck's constant. The fact that it is not zero ensures that uncertainty is a deep property of nature. Also, the fact that it has such a small value ensures that this uncertainty affects only objects at the quantum scale.

So, uncertainty and probability are necessary ingredients in the workings of the universe. It is not an anthropocentric phenomenon as we are just a part of it. Does anybody know why it has to be this way? Perhaps this is the way nature manages to achieve all the beautiful divergence we observe in it. This is probably the reason why she always has better ways to go about something than we can think off!!

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Very valuable explanation I think, +1. –  Jeroen Moons Jun 16 at 15:09
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