Cause of the Heisenberg Uncertainty Principle

Hi! So I know what the Uncertainty principle states and everything, but I can't find anywhere the "causes" of the uncertainty. Like, what is it exactly that "causes" the uncertainty. If it's not the measurement tools/technology, then what is it?? Please explain in terms of physical phenomena, not in terms of Fourier transforms that I wouldn't understand. Thanks!

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Borek
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I can be wrong, but I guess whoever answers the question gets an instant Nobel prize in physics.

Note, that many phenomena we can describe are not something that we know cause of - but we know they exist because we observe them in our experiments. We rarely know answer to the question "why", much more often we can describe "how".

the most simple explanation I have been given of it is to imagine measuring a particle's position & momentum. you have to bounce light off it to detect where it is and where it's going, but by bouncing the light off it you affect those things. so the observation affects the particle, hence the uncertainty in what you measure. scaling up to every day situations it doesn't apply because things are so heavy that a bouncing a photon off them doesn't make a significant difference. i could be wrong, i am certainly no expert in this but that makes sense to me.

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jtbell
Mentor
To get a qualitative illustration of the uncertainty principle:

Just add together a bunch of waves with slightly different wavelengths. The result forms a "packet" with a localized maximum. (Actually if you add a finite number of waves you get a series of packets, but just focus on one of them.)

Now add together a bunch of waves that are spread out over a larger range of wavelengths. The packet becomes narrower (in terms of position).

I've attached a picture from a quick spreadsheet I made. It shows graphs of the following two sums:

$$y_1 = \cos(9x) + \cos(9.2x) + ... + \cos(10.8x) + \cos(11x)$$

$$y_2 = \cos(8x) + \cos(8.4x) + ... + \cos(11.6x) + \cos(12x)$$

(The wavelengths of the waves go like $2\pi/9$, $2\pi/9.2$, etc.)

The second set of waves spans a range of wavelengths about twice as large as the first set, and the resulting packet is about half as wide in terms of position.

The wavelength is associated with momentum via de Broglie's formula $p = h/\lambda$. Actually the momentum is more directly associated with the numbers 9, 9.2, etc. which are the wavenumbers (k) of the waves: $p = hk/2\pi$.

To get a more precise mathematical statement of the relationship between the widths, we need to define what we mean by "width" of a packet, and use Fourier analysis.

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jtbell
Mentor
the most simple explanation I have been given of it is to imagine measuring a particle's position & momentum. you have to bounce light off it to detect where it is and where it's going, but by bouncing the light off it you affect those things. so the observation affects the particle, hence the uncertainty in what you measure.
No. The uncertainty principle is not connected with the measurement process. In the early days of QM, Heisenberg himself used your example as an illustration of his uncertainty principle, but later, after he and others learned more about QM, he disavowed this "Heisenberg microscope." Nevertheless, it lives on in many introductory treatments of QM.

No. The uncertainty principle is not connected with the measurement process. In the early days of QM, Heisenberg himself used your example as an illustration of his uncertainty principle, but later, after he and others learned more about QM, he disavowed this "Heisenberg microscope." Nevertheless, it lives on in many introductory treatments of QM.
should it really be getting mentioned in lectures anymore then? I first heard this a lecture 3 years ago and that's the image I've had of it in my head ever since.

Borek
Mentor
should it really be getting mentioned in lectures anymore then?
I guess that's an unfortunate example of the fact that physicist are susceptible to inertia like any other objects.

tom.stoer
The Heisenberg uncertainty principle can be derived as a general "geometric" property of operators (observables) and Hilbert spaces. Therefore it has a mathematical origin ...

... but of course that raises the question what "causes" physics to be based on Hilbert spaces ...

QM postulates that all observable quantities are operators acting on a Hilbert space. The uncertainty principle is a direct result of the commutation relationships of these operators and the Cauchy-Schwarz inequality.

If that's a little math-heavy for you, imagine the operators as being matrices. You should be familiar with the idea that if you have two matrices A and B, and you consider their products, then it's not the case in general that AB = BA (they do not commute). QM says that observables are like these matrices, they do not commute and from this the uncertainty principle arises as a mathematical consequence.

That this description is accurate is experimentally verified. As for why nature obeys this rule, I have no idea whatsoever.

The Heisenberg uncertainty principle can be derived as a general "geometric" property of operators (observables) and Hilbert spaces. Therefore it has a mathematical origin ...

... but of course that raises the question what "causes" physics to be based on Hilbert spaces ...

Agreed; it all seems to come down to observables and whether they commute or not, which can be shown mathematically...but why? One of the great philosophical questions of our time...(and the last...what...80 or so years?)

The Heisenberg uncertainty principle can be derived as a general "geometric" property of operators (observables) and Hilbert spaces. Therefore it has a mathematical origin ...

... but of course that raises the question what "causes" physics to be based on Hilbert spaces ...
Actually the uncertainty principle can be derived without making any reference to Hilbert Spaces. The roots of the uncertainty principle lie in the fact that quantum operators do not necessarily commute with each other. There is no analog of this fact in classical physics. Whenever you have two operators in QM that do not commute, there is a corresponding Uncertainty relationship for those two operators. Conversely, whenever you have have two commuting operators, there is no Uncertainty relationship that connects those two operators. This statement applies to any QM operator, whether or not they operate on states in a "Separable" Hilbert Space or some other more general space. So, the uncertainty principle is far more general than the assumption that the non-relativistic QM is based on Hilbert spaces. In fact for sufficiently relativistic motion, many of the QM assumptions break down, but the Uncertainty Principle does not. It survives in QFT. As far as I know, we have no extensions of the field theory (such as string theory, or quantum gravity) where you are able to circumvent the Uncertainty Principle.

tom.stoer
Actually the uncertainty principle can be derived without making any reference to Hilbert Spaces. ... It survives in QFT. As far as I know, we have no extensions of the field theory (such as string theory, or quantum gravity) where you are able to circumvent the Uncertainty Principle.
Even QFT and QG (especially LQG) can be formulated using separable Hilbert spaces.

There are attempts in string theory to modify or to "generalize" The uncertainty principle for x and p

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Please explain in terms of physical phenomena, not in terms of Fourier transforms that I wouldn't understand. Thanks!
is similar to the one that Richard Feynman addressed in http://www.youtube.com/watch?v=wMFPe-DwULM&feature=player_embedded". I guess Feynman would answer it similarly as he has answered the question about "Why magnets attract and repel?" ;-)

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what is it exactly that "causes" the uncertainty.
Your question is self-contradictory. If we knew the cause of the quantum uncertainty, then it won't be uncertainty anymore. The fact is that microworld behaves randomly, and there is no explanation of this randomness. But, perhaps there is no point in looking for the explanation. It makes sense to try to explain things that are regular, logical and predictable. Quantum behavior is not like that. It is unpredictable and random. So, it doesn't require explanation. For me that's the beauty of quantum mechanics.

Eugene.

tom.stoer
The Heisenberg uncertainty principle does not say that the quantum world is completely random. There is some regularity in this randomness; some well-defined rules which can be applied. One can even prepare coherent states in quantum optics minimizing the uncertainty reletaion. So from that perspective it may seem reasonable to ask for the cause of randomness in just the same way as one may ask for the cause of regularity in classical physics.

But I think that you cannot answer these questions in principle; that means that the answer in classical mechanics is neither better nor worse "known" than in quantum mechanics. These questions belong tometaphysics, not physics.

Look at a very simple equation like F=ma; can you explain what causes this equation? I don't think so. You cannot say what the reason behind this equation is; all we know is that it works.

Now going to the quantum world things don't become worse. There are different equations; they do no longer describe the same "reality" we are "familiar with", but again they "work" in some sense. But the reason behind the equations is unknwon again.

strangerep
Years ago, a lecturer gave the observation that noncommuting operations should
not be so strange or unexpected in the macroscopic world. There's plenty of
examples. e.g., having a shower and then getting dressed is not the same as
getting dressed and then having a shower. :-)

I think of the uncertainty principle as being caused by the wavelike nature of matter. You can understand this in an interesting way, by listening to some sound waves, http://scienceblogs.com/builtonfacts/2010/03/hearing_the_uncertainty_princi.php" [Broken].

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