6 ms·

The formula for pi buried in a hydrogen atom

119 points by silasrude 6y ago

segfaultbuserr 6y ago
Original paper: * Friedmann, T., & Hagen, C. R. (2015). Quantum mechanical derivation of the Wallis formula for π. Journal of Mathematical Physics https://sci-hub.tw/10.1063/1.4930800 https://sci-hub.tw/10.1063/1.4930800 The result didn't come from experimental works, it's a pure theoretical derivation for the sake of mathematical physics. > The existence of such a derivation indicates that there are striking connections between well-established physics and pure mathematics that are remarkably beautiful yet still to be discovered. The paper only has three pages - it set up a particular calculation on the energy levels of hydrogen and obtains a limit, thus recovering the Wallis formula for π. If you know QM (I don't), the paper may be a fun read, it's a short and understandable calculation.
- rrss 6y ago
  The preprint is on arxiv too: https://arxiv.org/pdf/1510.07813.pdf https://arxiv.org/pdf/1510.07813.pdf
  - JadeNB 6y ago
    Routine reminder: please link to arXiv abstract pages, rather than directly to PDFs. https://arxiv.org/abs/1510.07813 https://arxiv.org/abs/1510.07813
    - rrss 6y ago
      whoops, thanks for the reminder :)
- algon33 6y ago
  Thanks for the link. You guessed well, it is a fun read. Its quite satisfying you get the property from the rotational aspect of the QM.
doovd 6y ago
The formula is also buried in a simple circle.
- FigmentEngine 6y ago
  no that's tau
  - z3ncyberpunk 6y ago
    Tau is just pi * 2
    - opless 6y ago
      PI is Tau/2
      - perl4ever 6y ago
        Isn't there a page somewhere with a lot of physics formulas in terms of tau instead of pi for comparison? I'm not very math or physics literate, but it was interesting to contemplate.
- HPsquared 6y ago
  And a hydrogen atom is spherical... Coincidence?
geoffmunn 6y ago
I really love the idea that pure mathematics and nature are the same thing. On a philosophical level, our daily lives are the expression of the differences and inefficiencies of our systems compared to an optimal end-state. Also, sometimes I think that religion and science have more in common than people think.
- JadeNB 6y ago
  > I really love the idea that pure mathematics and nature are the same thing. On a philosophical level, our daily lives are the expression of the differences and inefficiencies of our systems compared to an optimal end-state. Also, sometimes I think that religion and science have more in common than people think. The first of these isn't a coincidence; pure mathematics evolved from applied mathematics, which originally was the only kind, and which was specifically designed to handle nature. It has since departed from those roots, but is still guided by human intuition and æsthetic sense, both of which are deeply shaped by our experiences in the natural world. (For a counter-argument from a much better source, see https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness... .)
- user234683 6y ago
  > I really love the idea that pure mathematics and nature are the same thing This might not be what you mean, but there is a philosophical position that holds that all mathematical structures exist, and that our universe is simply one of these structures: https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...
  - gqcwwjtg 6y ago
    If you assume some notion of an object's description "really existing", it's easy to then assume the Universe must have a zero information description, since otherwise the description would be "outside" the Universe. And then you've backed yourself into a corner: what could zero information describe that also contains our observed universe? All mathematical structures it is.
SonOfLilit 6y ago
This brings to mind Richard Hamming's wonderful essay "The Unreasonable Effectiveness of Mathematics", which really blew my mind ten years ago: "But if you do not like these two examples, let me turn to the most highly touted law of recent times, the uncertainty principle. It happens that recently I became involved in writing a book on Digital Filters [8] when I knew very little about the topic. As a result I early asked the question, "Why should I do all the analysis in terms of Fourier integrals? Why are they the natural tools for the problem?" I soon found out, as many of you already know, that the eigenfunctions of translation are the complex exponentials. If you want time invariance, and certainly physicists and engineers do (so that an experiment done today or tomorrow will give the same results), then you are led to these functions. Similarly, if you believe in linearity then they are again the eigenfunctions. In quantum mechanics the quantum states are absolutely additive; they are not just a convenient linear approximation. Thus the trigonometric functions are the eigenfunctions one needs in both digital filter theory and quantum mechanics, to name but two places. "Now when you use these eigenfunctions you are naturally led to representing various functions, first as a countable number and then as a non-countable number of them—namely, the Fourier series and the Fourier integral. Well, it is a theorem in the theory of Fourier integrals that the variability of the function multiplied by the variability of its transform exceeds a fixed constant, in one notation l/2∏. This says to me that in any linear, time invariant system you must find an uncertainty principle. The size of Planck's constant is a matter of the detailed identification of the variables with integrals, but the inequality must occur. "As another example of what has often been thought to be a physical discovery but which turns out to have been put in there by ourselves, I turn to the well-known fact that the distribution of physical constants is not uniform; rather the probability of a random physical constant having a leading digit of 1, 2, or 3 is approximately 60%, and of course the leading digits of 5, 6, 7, 8, and 9 occur in total only about 40% of the time. This distribution applies to many types of numbers, including the distribution of the coefficients of a power series having only one singularity on the circle of convergence. A close examination of this phenomenon shows that it is mainly an artifact of the way we use numbers. "Having given four widely different examples of nontrivial situations where it turns out that the original phenomenon arises from the mathematical tools we use and not from the real world, I am ready to strongly suggest that a lot of what we see comes from the glasses we put on. Of course this goes against much of what you have been taught, but consider the arguments carefully. You can say that it was the experiment that forced the model on us, but I suggest that the more you think about the four examples the more uncomfortable you are apt to become. They are not arbitrary theories that I have selected, but ones which are central to physics. http://inters.org/Hamming-Unreasonable-Effectiveness-Mathematics http://inters.org/Hamming-Unreasonable-Effectiveness-Mathema... I don't know any QM, but it does not surprise me in the least that, trying to calculate something about rotation, you end up with a formula for Pi. Is it there because of a connection to how the world works, or is it there because we happened to ask a question where two effects of the real world canceled each other out and all that remains is something about math? Sure, "no circles were involved", but rotation was involved, and it is just as Pi-heavy a model as anything involving circles. Also, QM is full of integrals of sines, for reasons explained in the quote above, which introduce a bunch of Pi scaling factors that have nothing to do with reality and everything to do with the tools we chose to model it with, etc'. (I'm not saying that the universe doesn't seem to have a lot of Pi factors in its machinery, just that this specific case doesn't sound like one of them, and that the distinction is interesting and mind blowing, to me at least).
jacinabox 6y ago
Clever of them to hide pi in there eh?
- irrational 6y ago
  Wasn't there something like this in the book Contact? Something hidden in pi?
tobrien6 6y ago
This is from 2015 FYI. Here's a more in-depth summary: https://www.forbes.com/sites/kevinknudson/2015/11/10/everything-you-ever-wanted-to-know-about-pi-part-2-a-new-proof-of-the-wallis-formula-via-physics/#3df5d3021150 https://www.forbes.com/sites/kevinknudson/2015/11/10/everyth...

119 points by silasrude 6y ago

Original paper: * Friedmann, T., & Hagen, C. R. (2015). Quantum mechanical derivation of the Wallis formula for π. Journal of Mathematical Physics https://sci-hub.tw/10.1063/1.4930800 https://sci-hub.tw/10.1063/1.4930800 The result didn't come from experimental works, it's a pure theoretical derivation for the sake of mathematical physics. > The existence of such a derivation indicates that there are striking connections between well-established physics and pure mathematics that are remarkably beautiful yet still to be discovered. The paper only has three pages - it set up a particular calculation on the energy levels of hydrogen and obtains a limit, thus recovering the Wallis formula for π. If you know QM (I don't), the paper may be a fun read, it's a short and understandable calculation.

rrss 6y ago
The preprint is on arxiv too: https://arxiv.org/pdf/1510.07813.pdf https://arxiv.org/pdf/1510.07813.pdf
- JadeNB 6y ago
  Routine reminder: please link to arXiv abstract pages, rather than directly to PDFs. https://arxiv.org/abs/1510.07813 https://arxiv.org/abs/1510.07813
  - rrss 6y ago
    whoops, thanks for the reminder :)
algon33 6y ago
Thanks for the link. You guessed well, it is a fun read. Its quite satisfying you get the property from the rotational aspect of the QM.

doovd 6y ago

The formula is also buried in a simple circle.

FigmentEngine 6y ago
no that's tau
- z3ncyberpunk 6y ago
  Tau is just pi * 2
  - opless 6y ago
    PI is Tau/2
    - perl4ever 6y ago
      Isn't there a page somewhere with a lot of physics formulas in terms of tau instead of pi for comparison? I'm not very math or physics literate, but it was interesting to contemplate.
HPsquared 6y ago
And a hydrogen atom is spherical... Coincidence?

geoffmunn 6y ago

I really love the idea that pure mathematics and nature are the same thing. On a philosophical level, our daily lives are the expression of the differences and inefficiencies of our systems compared to an optimal end-state. Also, sometimes I think that religion and science have more in common than people think.

JadeNB 6y ago
> I really love the idea that pure mathematics and nature are the same thing. On a philosophical level, our daily lives are the expression of the differences and inefficiencies of our systems compared to an optimal end-state. Also, sometimes I think that religion and science have more in common than people think. The first of these isn't a coincidence; pure mathematics evolved from applied mathematics, which originally was the only kind, and which was specifically designed to handle nature. It has since departed from those roots, but is still guided by human intuition and æsthetic sense, both of which are deeply shaped by our experiences in the natural world. (For a counter-argument from a much better source, see https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness... .)
user234683 6y ago
> I really love the idea that pure mathematics and nature are the same thing This might not be what you mean, but there is a philosophical position that holds that all mathematical structures exist, and that our universe is simply one of these structures: https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...
- gqcwwjtg 6y ago
  If you assume some notion of an object's description "really existing", it's easy to then assume the Universe must have a zero information description, since otherwise the description would be "outside" the Universe. And then you've backed yourself into a corner: what could zero information describe that also contains our observed universe? All mathematical structures it is.

SonOfLilit 6y ago

This brings to mind Richard Hamming's wonderful essay "The Unreasonable Effectiveness of Mathematics", which really blew my mind ten years ago: "But if you do not like these two examples, let me turn to the most highly touted law of recent times, the uncertainty principle. It happens that recently I became involved in writing a book on Digital Filters [8] when I knew very little about the topic. As a result I early asked the question, "Why should I do all the analysis in terms of Fourier integrals? Why are they the natural tools for the problem?" I soon found out, as many of you already know, that the eigenfunctions of translation are the complex exponentials. If you want time invariance, and certainly physicists and engineers do (so that an experiment done today or tomorrow will give the same results), then you are led to these functions. Similarly, if you believe in linearity then they are again the eigenfunctions. In quantum mechanics the quantum states are absolutely additive; they are not just a convenient linear approximation. Thus the trigonometric functions are the eigenfunctions one needs in both digital filter theory and quantum mechanics, to name but two places. "Now when you use these eigenfunctions you are naturally led to representing various functions, first as a countable number and then as a non-countable number of them—namely, the Fourier series and the Fourier integral. Well, it is a theorem in the theory of Fourier integrals that the variability of the function multiplied by the variability of its transform exceeds a fixed constant, in one notation l/2∏. This says to me that in any linear, time invariant system you must find an uncertainty principle. The size of Planck's constant is a matter of the detailed identification of the variables with integrals, but the inequality must occur. "As another example of what has often been thought to be a physical discovery but which turns out to have been put in there by ourselves, I turn to the well-known fact that the distribution of physical constants is not uniform; rather the probability of a random physical constant having a leading digit of 1, 2, or 3 is approximately 60%, and of course the leading digits of 5, 6, 7, 8, and 9 occur in total only about 40% of the time. This distribution applies to many types of numbers, including the distribution of the coefficients of a power series having only one singularity on the circle of convergence. A close examination of this phenomenon shows that it is mainly an artifact of the way we use numbers. "Having given four widely different examples of nontrivial situations where it turns out that the original phenomenon arises from the mathematical tools we use and not from the real world, I am ready to strongly suggest that a lot of what we see comes from the glasses we put on. Of course this goes against much of what you have been taught, but consider the arguments carefully. You can say that it was the experiment that forced the model on us, but I suggest that the more you think about the four examples the more uncomfortable you are apt to become. They are not arbitrary theories that I have selected, but ones which are central to physics. http://inters.org/Hamming-Unreasonable-Effectiveness-Mathematics http://inters.org/Hamming-Unreasonable-Effectiveness-Mathema... I don't know any QM, but it does not surprise me in the least that, trying to calculate something about rotation, you end up with a formula for Pi. Is it there because of a connection to how the world works, or is it there because we happened to ask a question where two effects of the real world canceled each other out and all that remains is something about math? Sure, "no circles were involved", but rotation was involved, and it is just as Pi-heavy a model as anything involving circles. Also, QM is full of integrals of sines, for reasons explained in the quote above, which introduce a bunch of Pi scaling factors that have nothing to do with reality and everything to do with the tools we chose to model it with, etc'. (I'm not saying that the universe doesn't seem to have a lot of Pi factors in its machinery, just that this specific case doesn't sound like one of them, and that the distinction is interesting and mind blowing, to me at least).

jacinabox 6y ago

Clever of them to hide pi in there eh?

irrational 6y ago
Wasn't there something like this in the book Contact? Something hidden in pi?

tobrien6 6y ago

This is from 2015 FYI. Here's a more in-depth summary: https://www.forbes.com/sites/kevinknudson/2015/11/10/everything-you-ever-wanted-to-know-about-pi-part-2-a-new-proof-of-the-wallis-formula-via-physics/#3df5d3021150 https://www.forbes.com/sites/kevinknudson/2015/11/10/everyth...