After validating my basic approach to solving Schrödinger’s equation by discovering a new pseudo-solution to hydrogen, I spent several months developing code that can handle helium’s ground state using parallelization. The code works, but I estimate that it would require $10,000 for electricity and a year of computation to check helium ansatz 16.6, with no guarantee of success, as I have no theory to predict which ansatz is required. The calculation could also be run on AWS for about $250,000. The code is not terribly efficient, so another idea is to optimize the Gröbner basis code, which should be more efficient. Another possibility is to study the problem theoretically in order to predict which ansatzen are required to find the solution.
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:
![\[\Psi = J_0(2\sqrt{x+r})\]](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!
Continue reading “A New Pseudo-Solution of Hydrogen”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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Resolution of Singularities
Brent’s math lecture for Catholic University’s weekly seminar.
Sage 9: Function Fields
A twenty minute overview of some of the new features in Sage 9.
The Facebook Integral
Facebook has developed a neural network that can solve some integrals that Mathematica can not. In the first video, I explain how to use free software products (Axiom and Sage) to solve this integral. In the second video, I go into more detail about Robert Risch’s theorem and how it applies to the Facebook integral.
Sage 9: Multivariate Factorization over Q̅
Exact Computation of Algebraic Varieties using Numerical Techniques
(graduate seminar lecture at Catholic University)
Ke Alaka’i
I spent several months in Laie, Hawaii, where the local college magazine gave me some very good press.
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