## A New Pseudo-Solution of Hydrogen

An January 2023, I discovered a previously unknown solution to the simplest Schrödinger equation for the hydrogen atom. It’s my only truly new mathematical discovery to date, so I like to keep it pinned at the top of the blog.

It turns out that this wavefunction:

where J₀ is the ordinary Bessel function J₀, solves the Schrödinger equation for hydrogen:

with E=0.

It does not, however, satisfy the global integrability condition required for it to be a valid wavefunction:

Therefore, it is only a pseudo-solution, since it satisfied the differential equation but is not a valid wavefunction.

What I find 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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