The formula for pi buried in a hydrogen atom
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* 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
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.
Its quite satisfying you get the property from the rotational aspect of the QM.
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... .)
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_hypothes...
Pure maths (as a tendency, not strictly) focuses on formal systems that are on the consistent end of the spectrum (as those are the ones that can be used to build more maths on top of). Nature probably doesn't (or needn't) have this restriction/tendency! Formally consistent might be higher performing in many cases (more stable over long time-spans, like dna), but it is not the end of the story as to what is ultimately possible in rule-following systems! Any sense of determinism demands this, even if the system makes its own rules as the universe/reality - the ultimate all, with nothing outside of it - must. Whether this fully gives way to a properly random (like an oracle of rand) system at the quantum level is still an open question.
It is possible to expand ones view of nature/ultimate-reality to include other rule-systems (such as including inconsistent or entirely 'complete' systems.) This can have more coverage (completeness) of any particular model, or even be totally incomplete and inconsistent! These 'degenerate' (according to maths) formal systems might sometimes be the ones that happen to work in nature/reality. Once you do this extension, now you're talking about computation, admitting a broader class of rule-following than pure maths. You can argue that the two are equivalent because sure, anything can be translated between them - i'm more talking about tendencies here. The computational universe includes lots of formal systems that are abhorrent or impractical to mathematical understanding, so mathematicians avoid them, but nonetheless they are still rule-following! Nature/evolution probably uses/finds those rules where they are the best performing solution (through its exploration of possibilities).
As to 'optimal end-states', I'm afraid you've lost me there. I gave up teleology (the idea that the universe or nature or evolution have any obvious goals) a long time ago, how could anything like that (high level) feed-in to reality? There is no hint of this in all of science. What self-organises and self-replicates is what persists. This self-support might even reach down into physics!
To me, religion is nearly the opposite to all of this: An explanatory system that says explanation-of-reality='fixed string' with no update method available. Very incomplete (unless ridiculously long, which current examples of religion are not) and inconsistent: The fixed string might assign True and False at once, or neither - to new knowledge that falls outside the original scope of its meaning.
So no, I don't think that this pi-result suggests that religion and science have a commonality here. Pi is awesome (even though very constructible) and we should probably expect to find it [0], especially in places like this (pure QM) that involve geometry.
"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-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).
I had an EE graduate degree (systems, dsp, control...) and returned to school for a physical chemistry graduate degree. Initially was nervous about my first QM class but as Hamming noted so much of the math is similar.
This is a part of the arguments used both pro/con of Tau over Pi. Constants that start with 6 are "weird" because they are uncommon. They are uncommon because "we" like to just halve them and use constants that start with 3 and lots of factors of two when working with them. It's a weirdness entirely of mathematics' own creation. Neither is more "fundamental", people just likes 3s sometimes too much.
In any unit you choose, about 30% of rivers on Earth and about 30% of public company market caps will start with a 1. This has nothing to do with people's preferences and everything to do with scale invariance.