Friday, February 3, 2012

How do you get spherical symmetry from all that junk?

For the last couple of posts, I've been all excited because I say I've written a spherically symmetric wave function for two electrons bound to a central charge. I know you must be thinking: the guy is nuts. Look at the wave function, and it's got all kinds of stuff in theta and phi; it's got one electron leaning this way and another one leaning that way. How can you possibly get spherical symmetry from all that junk?

It's a good question and it actually goes to the heart of some very deep issues in quantum mechanics. When Schroedinger invented the wave function in 1926, the big motivation was to put the mysteries of quantum theory firmly in the realm of causal interactions taking place in real space-time. It was a huge success for the hydrogen atom. But almost immediately, everyone who was anybody understood that the obvious generalization to multi-particle systems took you into 3n-dimensional phase space, where n is the number of particles. So the helium atom has to be solved in six-dimensional phase space, not in "real" space-time. It's a problem.

What people seem to forget is that these n-dimensional solutions, although mathematically correct, seem to overspecify things to a certain extent. The obvious example is the phase of the wave function: clearly, you can multiply a solution by any arbitrary phase and it's still the same physical solution. But the bigger problem is the whole business of the particles. The generalized n-dimensional Schroedinger equation requires you to describe the physics in terms of particles being here and there; but in reality, you are not allowed to distinguish between different particles. So there is something called "symmetrization" you have to do with your solution, which means your saying "A is here and B is there OR B is here and A is there...". Or something like that.

But in our deepest understanding of reality, we can't really believe that electrons A and B have independent existence...otherwise, why would we have this whole structure for interchanging them and getting the same physics? And yet we don't seem to know how to solve for a system of electrons without starting off, in our human perspective, by labelling them A, B, C etc, as though they are independent, distinguishable entites; and then, after the work is all done, we "symmetrize" the solution, effectively robbing the electrons of their independent existence. We just don't know how to get there without without going through that whole song-and-dance.

In the case of proto-helium, I have one electron biased towards the Northern Hemisphere, and the other one biased towards the south. The angular dependences of charge density are cos-squared for the first one, as measured from the North Pole, and sin-squared for the other one: obviously the charge density is uniform over the sphere. Similarly, the spin orientations add up so that the first one is pointing everywhere radially outwards, and the second one everywhere radially inwards: the total spin must therefore be everywhere zero. It's almost obvious that if I had started with one electron in the eastern hemisphere and the other in the west, I would have gotten the same physical solution in the end. The whole north-south orientation was just a convenient fiction that helped us do our bookkeeping.

The big question is: if every physical manifestation of our solution is spherically symmetrical, then why does the equation have so much intricate structure built into it? The answer can only be that there is a huge redundancy built into the n-dimensional Schroedinger equation, and it flows from the premise that we must start off with independent distinguishable particles. If, in the end, we get a physical solution which contains no trace of those independent entities, we have to ask ourselves: did those independent electrons ever really exist?

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