## Sunday, May 1, 2011

### Hydrogen Spin Singlet State

I noticed a few weeks ago that there is a spin singlet state for hydrogen. Not for a single hydrogen atom, but for two isolated hydrogen atoms. I've been second-guessing myself wondering if it could be, but I keep turning it over and over and it comes out right. I'm going to redo the calculation here and see how it goes.

The "conventional" way to describe the spin of two separate hydrogen atoms is to identify four possibilities: up/up, down/down, up/down, and down/up. These are oversimplifications that obscure the fact that the two electrons must be indistinguishable. For each state, we have to consider that electron A might be on the left and B on the right, or vise-versa. So, for example, the up/up state is really (A-up/B-up) or (B-up/A-up). If we replace the "or" by a plus sign, we get a problem because if you reverse the roles of A and B you get back the same state you started with; and for electrons, reversing A and B must always return the same state with a minus sign. We can fix this by considering the state

(A-up/B-down) - (B-up/A-down)

This gives the desired result when you swap A and B. In fact, all four states can be fixed in exactly this way; so with the understanding that this is what we really mean, we will continue to identify our four states with the simplified notation:

1. up/up
2. down/down
3. up/down
4. down/up

Now there is something crazy about this system in that it appears the two separate hydrogen atoms can have their spins either aligned, or anti-aligned, and nothing in between. This is physically nonsense, and the answer must be that we can take superpositions of these four states to give us any desired combination of spin orientations. We'll come back to that in a bit, but first we want to recongnize the very special thing that happens when we take the difference of states 3 and 4:

A(up)B(down) - A(down)B(up)

It's called the singlet state and it is very different from all other combinations of states in that it is spherically symmetric with regard to spin. We can demonstrate this by doing a change of basis and writing it down as it would appear if rotated 90 degrees. You know that spin up becomes (up+down) when viewed sideways, and spin down becomes (up-down). Let's put these valuses into the expression and evaluate them:

A(up+down)*B(up-down) - A(up-down)*B(up+down)

Remember that order is important here, so we get eight terms:

A(up)B(up) - A(up)B(down) +A(down)B(up) - A(down)B(down) -A(up)B(up) - A(up)B(down) + A(down)B(up) + A(down)B(down)

Collecting terms and renormalizing, we get the result:

-A(up)B(down) + A(down)B(up)

This is the same expression we started out with (except for a minus sign). It means the spin state looks the same if you rotate 90 degrees. Since 90 degrees was just an arbitrary trial, we can expect to get the same spin for any rotation. If this is so, then the spin state is spherically symmetric, and the only way this can be is if the spin is everywhere identically zero. The spin singlet state.

As I pointed out last time the subject came up, it seems very odd that by looking at two hydrogen atoms in combination, we can identify a behavior that is almost unthinkable if we consider just a single hydrogen atom in isolation.