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Friday, March 5, 2010

Example Of A Crazy Model: The Path Integral.

I recently discussed how it may be that mathematical models that describe our universe may or may not be what is actually going on but: it's okay since these models have predictive power.

For example, take the famous path integral.

Remember the double slit experiment shown in the picture below.  If you ask: what is the probability that an elementary particle released at point s, it ends up at point o?  It turns out, being non-technical here, to get the right answer you have to assume the particle to go through both holes together.


Now, run the same experiment where now you have two screens with multiple slits.( See picture below)  Again, to get the right answer, you need to assume the particle, in some sense, travels from s to o in every path possible.
No, add in infinite number of screens and drill into them an infinite number of holes.  What do you get?  Free space!  Yet surprisingly, you get the right quantum physics if you assume the particle still has to travel through all the infinite holes of the infinite screens: Ie, you assume the particle moves from point s to o taking every path possible in free space.

Now, though I didn't state all the details 100% technically correct, the basic intuitive idea of what is going on is still correct.  The math we use is suggests the particles take, in some sense, every possible path.

Now, is this really what is going on, or is this just a model?  I'm guessing it is just a model.  However, it's okay since it has significant predictive power.

Nevertheless, the question is always going to pester me: why do such bizarre models give such amazing answers?!?!  Nature is very interesting indeed.

(These images were taken from Quantum Field Theory In A Nutshell By A. Zee. This book has an amazing section on path integrals.)

12 comments:

  1. My research group has a curriculum published by Wiley called Tutorials in Introductory Physics, and we make heavy use of models. In the tutorial entitled "A Model for Single-Slit Diffraction," for example, we use ideas developed previously for multiple-slit interference to build a model in which a single slit is imagined to be a huge number of tiny slits (essentially with the distance between slit centers equal to the tiny slit widths). It's absurd, but it gives the right answers. How wonderful!

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  2. Yeah, stuff like this is really cool. I had a professor like to always say when he discussed things like this: "Nature is weird". And so it is, and yet so interesting.

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  3. And thankfully so. If nature weren't so weird, we'd all be out of business.

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  4. Your right Bill. I remember we had the Nobel Prize winning physicist Leon Lederman visit our quantum mechanics class (Weren't you there Bill?) at BYU who said at one point "It would be great if the LHC discovers something crazy. Well all have jobs for sure." (paraphrasing)

    And so it is. :)

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  5. Thought the slit concept was more than just a model. Don't the interference and entanglement experiments prove the model correctly reflects nature?

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  6. " Don't the interference and entanglement experiments prove the model correctly reflects nature?"

    Yes, that's my point. The model correctly 100% predicts nature.

    But the same model mathematically suggests when a particle moves from point A to be it travels in all possible paths. Does this really happen? But, since it's measurable predictions 100% fit nature, the model is none-the-less good.

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  7. Anyways, I am being *way* too philosophical here.

    I'm only trying to illustrate how it is hard to know if the model is right because this is how nature really is or if it is right just because it is mathematically equivalent to what is actually going on.

    Either way its predictions are always observed to be correct.

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  8. Well I like to think that particles do take every pathway. It gives the Universe a higher Star Trek correlation factor. =:)

    btw, a good pop sci book on the topic is:
    'Entanglement' by Amir D. Aczel

    I recommend all of his books.

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  9. Stan, thanks for the book link. I'm interested in learning more about entanglement.

    And you know, the universe may really be this way.

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  10. In “Reasoning and the Logic of Things”, the Cambridge Conferences, lectures of 1898 by Charles Sanders Peirce ; it is possible to read :
    “…It seems that we are reduced to this alternative : either we have to do a very large generalization about the character of the ways of nature ; it can at least say to us that it is better to try such a theory of molecules and ether rather than such an other ; or…”
    “…About these explanations proposed by the physicists for the irreversible phenomenons by means of the doctrine of random applied to some billions of molecules, I accept them fully, as being one of the most beautiful successes of science.”
    I think that there is a connection.

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  11. I did translate so I hope it is correct.

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