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!

Here’s a preprint of a paper that I’m working on to explain the result:

3 Replies to “A New Solution of Hydrogen”

  1. I went out to UC Berkeley two weeks ago and spoke with a number of people in the math department. All of them gave me good advice.

    Prof. Evans wondered how my result could be analyzed with Peter Olver’s theory (GTM 107); it’s a good question that I need to think about some more. Prof. Grunbaum suggested looking in the Pauling and Wilson quantum mechanics book, which I did (the formula wasn’t there). Thomas Browning and a group of grad students suggested putting a paper up on arxiv.org

    Turns out that putting a paper up on arxiv is difficult for people outside of academia. You have to find someone to endorse you, and none of the Ph.D’s I know are considered endorsers for the mathematical physics archive. Paul Kainen, for example, could endorse me in combinatorics (he’s a graph theory specialist), but that’s the wrong archive.

    My plan was to put a draft up on arxiv and revise it every few days until it’s ready. Instead, maybe I’ll just revise it here on freesoft.org until I find someone to endorse me on arxiv.org

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