A New Solution of Hydrogen

About a month ago, I discovered a previously unknown solution to the simplest Schrödinger equation for the hydrogen atom.

It turns out that this wavefunction:

    \[\Psi = J_0(2\sqrt{x+r})\]

where J₀ is the ordinary Bessel function J₀, solves this equation:

    \[-\frac{1}{2}\nabla^2\Psi - \frac{1}{r} \Psi = E\Psi\]

with E=0.

What I find more surprising about the result is that it solves one of the best known equations in mathematical physics, yet has apparently remained undiscovered for over a hundred years!

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The Soul of Quantum Mechanics

The soul of quantum mechanics, to me, is the complex number system.

When we look around us, we see what mathematicians call “real three-dimensional space”, at least that’s what it looks like. Who knows what it actually is, but that’s what it looks like.

In much the same manner, quantum mechanics looks like complex space, of some kind. The more I study quantum mechanics, the more it looks to me like complex numbers, not real numbers, are the basic kind of number system that it’s made of.
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The Facebook Integral

Facebook has developed a neural network that can solve some integrals that Mathematica can not. In the first video, I explain how to use free software products (Axiom and Sage) to solve this integral. In the second video, I go into more detail about Robert Risch’s theorem and how it applies to the Facebook integral.