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:
![\[\Psi = J_0(2\sqrt{x+r})\]](https://quicklatex.com/cache3/74/ql_4a719605c50267d160a2c26cf6cfe574_l3.png "Rendered by QuickLaTeX.com")
where J₀ is the ordinary Bessel function J₀, solves the Schrödinger equation for hydrogen:
![\[-\frac{1}{2}\nabla^2\Psi - \frac{1}{r} \Psi = E\Psi\]](https://quicklatex.com/cache3/c2/ql_c88e1d19064771f724a700e207df53c2_l3.png "Rendered by QuickLaTeX.com")
with E=0.
It does not, however, satisfy the global integrability condition required for it to be a valid wavefunction:
![\[\int|\Psi|^2 < \infty\]](https://quicklatex.com/cache3/ea/ql_a0258354f17354d56ccf53e0f9df03ea_l3.png "Rendered by QuickLaTeX.com")
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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