Sunday, February 5, 2012

Proto-Helium: What is it good for?

Last week, with great fanfare, I announced the solution of the six-dimensional Schroedinger equation for Proto-Helium, a mythical atom composed of two electrons and most of a proton, such that the species is just marginally stable with respect to loss of the second electron. It's undoubtedly a picturesque idea; even if my partial solution, such as it is, happens to be incorrect, it still illustrates the necessary existence of a mathematical solution fulfilling those particular specifications. But what can we do with proto-helium? Does it hold the key for understanding the solution of the helium atom?

Sadly, after all that, I still don't know where to go from here. I don't even know how to write the radial dependence of the wave function for proto-helium, but that doesn't even bother me so much. The worst of it is I don't know how to take this idea and leverage the solution for helium. I've laid out the isoelectronic series of helium from one end to another, starting with my mythical proto-helium at one end and ending with the limit of infinite nuclear charge; and I've got plausible expressions for the solutions to both those limiting cases in terms of simple product functions or sums thereof. What I still don't know how to do is to blend from limit to the other.

And it's not for lack of trying. I've been playing around with all kinds of superpositions, but none of them are helpful.  I know a lot of things that won't work, but I haven't come up with anything that does. That doesn't necessarily mean I've hit a dead end: it just means I don't know where to go yet. I might figure it out tomorrow, or I might never figure it out. That's how it goes in physics.

There is one small advantage I've gleaned from inventing proto-helium. I've shown myself how to write a spherically symmetric charge distribution by starting with an arbitrary axis of separation for the two charges. That's a non-trivial problem: you always have to start that way, but the question is how do you end up symmetrizing it? Do you need an infinite distribution of such solutions taken over all possible orientations? What proto-helium shows me is that you don't: you can satisfy your need for spherical symmetry with a superposition of as few as four simple product functions. I think that's worth something: I just still don't know exactly what to do with it.

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