Wednesday, February 10, 2010

Something I just figured out yesterday

People go on and on about the deep mysteries of quantum mechanics. It seems like the double slit experiment is a huge mind-bender for people. Not for me. I can do the double-slit calculations and they make sense. Basically I can handle stuff in quantum mechanics when there is one electron at a time. It's when you add the second electron that it gets messed up.

I finally decided that it should be possible to work some things out by taking the very simplest two-electron case. No, it's not the Helium atom, although that's also an important one. No, it's not the Hydrogen molecule either. No, it's not the scattering of one electron by another. And it's not even the infinite potential well with two electrons in it. I can't do any of these problems.

What I've tried to do is the very simplest possible case. It's the case of two isolated hydrogen atoms, treated as a single system with two electrons. Now why would you do it that way? You can solve for a single hydrogen atom, and then if you have a second one, you've already solved it. It's not exactly easy to solve for one electron in the sense that you've got to differentiate stuff in polar coordinates, but it's at least straighforward.

But that's why I chose to do it as a two-electron problem. You just get the familiar solution centered around each atom, and then combine them into symmetric and antisymmetric combinations. The symmetric state is the lowest energy.

Now bring in the electrons. You can put the first electron in that ground state and it is then shared between the two atoms. If you want to localize the electron at one atom, you have to use the combination of symmetric plus antisymmetric. But since the energies of these two states are slightly different, they gradually go out of phase with each other and eventually you find the electron at the other atom, even if that other atom is quite far away. That's what they call tunneling.

What gets interesting is the second electron. Forget everything you heard about two electrons being able to occupy the same state if they have the opposite spin: technically that's correct, but it fools people into thinking they can just drop two electrons into the states they've already calculated. While that would give you two nice hydrogen atoms at first glance, it's not really correct. It ignores the interaction energy between the two electrons.

If you've ever glanced at the solution for the helium atom it starts by writing the six-dimensional Schroedinger equation which basically has terms for the potential and kinetic energy similar to the hydrogen atom, plus a new term in 1/(r1-r2) which is the interaction potential. Now, what occurs to me in the case of two hydrogen atoms, is that it's exactly the same equation. The way nature solves this equation in practise is that it anti-correlates the two electrons by putting one here whenever the other one is there, so the interaction term basically gives you zero and you get the familiar solutions. But it occurs to me: why can't you let the hydrogen atoms solve this problem exactly the same way that helium does? In other words, take the "known" solution for helium - well, you can't easily write it down but somehow nature "knows" a solution" - and scale it by a factor of 2 on length so it fits the hydrogen atom. How can this not possibly be a correct solution of the Schroedinger equation? True, it's not the lowest energy solution, but it's still a solution.

So each hydrogen atom becomes a mini-helium, with two half-electrons vying for the same space. The energy is exaclty one-quarter of the helium ground state energy of -76 eV, which comes to something like -9.5 eV per electron. Of course, you can't ionize it with 9.5 eV, you need to use 19 eV because its a coupled system and you have to ionize both at the same time. But regardless, you've got a mathematical state which it seems to me should have some physical consequences.

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