{"id":434,"date":"2023-02-28T15:11:11","date_gmt":"2023-02-28T20:11:11","guid":{"rendered":"https:\/\/www.freesoft.org\/blogs\/soapbox\/?p=434"},"modified":"2025-06-11T23:16:43","modified_gmt":"2025-06-12T03:16:43","slug":"a-new-solution-of-hydrogen","status":"publish","type":"post","link":"https:\/\/www.freesoft.org\/blogs\/soapbox\/a-new-solution-of-hydrogen\/","title":{"rendered":"A New Pseudo-Solution of Hydrogen"},"content":{"rendered":"\n

A New Pseudo-Solution of Hydrogen<\/h2>\n\n\n\n

An January 2023, I discovered a previously unknown solution to the simplest Schr\u00f6dinger 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.<\/p>\n\n\n\n

It turns out that this wavefunction:<\/p>\n\n\n

  <\/span>   <\/span>\"\[\Psi = J_0(2\sqrt{x+r})\]\"<\/p>\n\n\n\n

where J\u2080 is the ordinary Bessel function J\u2080, solves the Schr\u00f6dinger equation for hydrogen:<\/p>\n\n\n

  <\/span>   <\/span>\"\[-\frac{1}{2}\nabla^2\Psi - \frac{1}{r} \Psi = E\Psi\]\"<\/p>\n\n\n\n

with E=0.<\/p>\n\n\n\n

It does not, however, satisfy the global integrability condition required for it to be a valid wavefunction:<\/p>\n\n\n

  <\/span>   <\/span>\"\[\int|\Psi|^2 < \infty\]\"<\/p>\n\n\n\n

Therefore, it is only a pseudo-solution, since it satisfied the differential equation but is not a valid wavefunction.<\/p>\n\n\n\n

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!<\/p>\n\n\n\n\n\n\n\n

Here’s a preprint of a paper that I’m working on to explain the result:<\/p>\n\n\n\n