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A New Pseudo-Solution of Hydrogen

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:

    \[\Psi = J_0(2\sqrt{x+r})\]

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

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

with E=0.

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

    \[\int|\Psi|^2 < \infty\]

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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U.S. Patent 11,669,350

Last year, I was awarded a U.S. patent for the remote desktop video conferencing system that I developed, collaborate.

Collaborate is, and will remain, open source and free to use. The purpose of the patent is only to claim my legal rights as the inventor; I have no intention of trying to monopolize the use of this technology.

Collaborate is hosted on a github repository.

11669350Download
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Baccala v. VEC

Last week I filed a lawsuit against the Virginia Employment Commission.

They seem to have developed an illegal administrative procedure called “Vacate” and have used it twice on my claims for unemployment benefits. This goes back to the summer of 2020. I fought them in federal court in 2022 and had them in state court earlier this year (2023). I had a lot of trouble finding a lawyer to take the case and have been representing myself pro se.

Update: I lost the case.

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The A.I. Revolutions

The recent surge of interest in Large Language Models (LLMs) like GPT-4 has motivated me to set aside my mathematical research for a time and focus instead on A.I. The current LLM revolution does not seem to be a passing fad.

I’m predicting not one, but three A.I. revolutions that I expect to follow in fairly short order:

  • The Large Language Model Revolution
  • The Machine Vision Revolution
  • The Personal Robotics Revolution
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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 Golden Age of Computing

Ah, the Golden Age of Computing!

It isn’t here yet.

Maybe a hundred years in the future.

I’ll be nice.

If you read this then, take joy! You’ve got a wonderful digital assistant that’s an expert draftsman, chessplayer and trivia champion, runs a tight, secure network presence for you, is always letting you know the latest interesting tidbits on the news feed or the personal side, sigh…

For now, I’ve got to deal with nginx configuration files.

When I got up this morning the network was down. I had moved a laptop yesterday from wired to wireless and back, it didn’t come back right and its used as a router because the real router doesn’t have a wifi interface. Probably still not right, but I got the power turned on to osito. It took a while.

Jotted some things down in my diary, took a smoke, had something to eat, back to work.

Now I’m trying to re-install collaborate. Remove the development version. Install the distributed version. Why is it so slow? Dig through Google pages because I can’t remember the name of the debugging program – that’s it! iftop! Why is it connecting secure? Why is my website so slow?

Oh, crap, my website is down (again). D—. Dig through the docs and keys to figure how to get into the aws console. Oh, it looks fine. Wait a minute, that was just a typo on my part, logging in to the wrong machine. False alarm.

Finished the install, but it’s broken. Stare at its logs. bbb-html5 didn’t install right. nginx couldn’t start. Check the nginx logs. Why didn’t that file delete? It was part of the removed package. Check the package versions. Yup, we got the right package, but that file shouldn’t be there. Who knows why it wasn’t removed. Remove it by hand. Complete the install.

OK, now I can connect, but no audio. Seen this plenty, just down and up the videoconferencing server. OK, now it works with audio!

It’s 3:30 in the afternoon. I just finished my first task for the day: install the currently distributed version of collaborate on osito.

To those of you reading this in the Golden Age of Computing, take joy!

We in the early twenty first century see their potential, but waste so many hours dealing with all these bugs, the lack of natural language recognition, and the bugs, the bugs, the bugs!

In the Golden Age of Computing, you’ll be able to get some work done.