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-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).