I’m sure you’re
all eagerly awaiting the conclusion of the story, where I write out the full
equation for the wave function of proto-helium, that mythical atom consisting
of two electrons and a fractional nuclear charge, which is just marginally
stable with regard to loss of the second electron. At the same time, I can
almost hear the naysayers scoffing at me, because I’ve heard it before: “So

*what*!! Who cares if you can calculate the wave function of an atom that doesn’t even exist. Don’t you know there is no such thing as nine-tenths of a proton?”
I know that’s
what people are saying because people have said that kind of thing to me all my
life. Well, I’m going to give the scoffers more ammunition today because I’m
not going to write out any more mathematical equations than I’ve already
written. The fact is, I don’t know what the radial portion of the function
looks like because I haven’t actually solved for it: to which I’m sure the
scoffers will howl out gleefully: “Then you have

*nothing*!!!”. Well, let my readers be the judge of that. I have the angular part of the function, which I think is still more than nothing. The question is: have I got it right?
When I solved
for the two-electron potential well, I did it by taking the simple product
function solution and adding a bit of second harmonic. I added it with opposite
phases to electrons A and B, which had the net effect of pushing the charge
distributions a little bit apart. This was the graph of the two wave functions:

All this
overhead is what makes the helium atom so very difficult. The one-dimensional
two-electron well must ultimately be solved in two-dimensional phase space; and
as I have written it out above, it may be most simply expressed as the sum of no less than four
simple product functions. My proto-helium atom will be very similar: the full
solution must exist in six-dimensional phase space, but in can be most simply
expressed as the sum of four simple product functions in ordinary
three-dimensional space. Of course, every wave function in 3-d space can
ultimately be expressed as the sum of simple product functions, but how many
functions does it take? I don’t know how to do it for helium, and I suspect it
might not be so simple. That’s why it’s so useful to work with the limiting
cases, such as proto-helium.

The nice thing
about simple product functions is that they retain a certain degree of physical
reality that gets lost when we make the transition into
six-dimensional phase space. If we look back at the one-dimensional potential
well, we can see that all the real physics, from a descriptive point of view,
takes place in the very first iteration of the solution, when we simply allowed
the two charges to push each other a little bit apart. Proto-helium is very
much the same: we have regular helium, where the electrons are almost on top of
each other; and as we weaken the central attraction of the nucleus, the charges
tend to move apart, until in the limiting case, their angular distribution follows this marvelous function which I would like to name after myself:
but since there is already a Green’s Function, I’m going to call this one “Marty’s
Function”:

In proto-helium, the angular distribtuion of the first electron is given by this function, and the second electron gets the same function except turned upside down. So one electron leans towards the northern hemisphere, the other one towards the southern. You then just take the simple product of those two functions. The marvellous thing is that if you carefully examine the distribution of charge and spin, you get a perfectly uniform charge distribution over the surface of the sphere, and the spin vector points everywhere radially outwards. So the function is spherically symmetric.But we're not done yet. Just like the two-electron well, we haven't properly symmetrized the function with respect to exchange of electrons. And just like the two-electron well, the full solution will be the sum of four simple-product functions identical to the one we've already described, except with spins this way and that. But the essential physics is to be found in the description we've already made so far.

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