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One dimension in position representation:
In three dimensions In position representation, we define it by using the gradient, and so we see that
Video 1.
Position and Momentum from Wavefunctions by Faculty of Khan (2018)
Source. Proves in detail why the momentum operator is . The starting point is determining the time derivative of the expectation value of the position operator.
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Music notation engine. domain-specific language input. The LaTeX of music.
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Dominating awesome list: https-github-com-nodiscc-awesome-linuxaudio
A large part of these projects are on SourceForge as of 2020, it is scary. They just die.
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Consider this is a study in failed computational number theory.
The approximation converges really slowly, and we can't easy go far enough to see that the ration converges to 1 with only awk and primes:
sudo apt intsall bsdgames
cd prime-number-theorem
./main.py 100000000
Runs in 30 minutes tested on Ubuntu 22.10 and P51, producing:
Figure 1.
Linear vs approximation plot
. and are added to give a better sense of scale. is too close to 0 and not visible, and the approximation almost overlaps entirely with .
Figure 2.
. It is clear that the difference diverges, albeit very slowly.
Figure 3.
. We just don't have enough points to clearly see that it is converging to 1.0, the convergence truly is very slow. The logarithm integral approximation is much much better, but we can't calculate it in awk, sadface.
But looking at: en.wikipedia.org/wiki/File:Prime_number_theorem_ratio_convergence.svg we see that it takes way longer to get closer to 1, even at it is still not super close. Inspecting the code there we see:
(* Supplement with larger known PrimePi values that are too large for \
Mathematica to compute *)
LargePiPrime = {{10^13, 346065536839}, {10^14, 3204941750802}, {10^15,
     29844570422669}, {10^16, 279238341033925}, {10^17,
    2623557157654233}, {10^18, 24739954287740860}, {10^19,
    234057667276344607}, {10^20, 2220819602560918840}, {10^21,
    21127269486018731928}, {10^22, 201467286689315906290}, {10^23,
    1925320391606803968923}, {10^24, 18435599767349200867866}};
so OK, it is not something doable on a personal computer just like that.
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