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shminux comments on So you think you understand Quantum Mechanics - Less Wrong
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Your MWI analysis is close to the mark. One thing that is not quite right is multiple states inside each term. Note that once an atom interacts with the screen, it no longer has a definite spin or even position. It becomes a part of the blob on screen, entangled with the atoms around it. Thus the interaction is better described as
blank screen(Awave packet moving forward and upatom spin up + Bwave packet moving forward and downatom spin down) -> Ascreen with top mark + Bscreen with bottom mark.
The atom state is buried somewhere inside the mark on screen.
The observer's interaction with the screen is similarly simplified:
You(Ascreen with top mark + Bscreen with bottom mark) -> AYou-topscreen with top mark+BYou-bottom*screen with bottom mark.
Note that you still have to apply the Born rule to calculate the prior probabilities of You becoming You-top and You-bottom.
* any potential double entendre is completely accidental.
I don't follow, on position. The atom had a distribution over position which was a smooth wave while it was in transit, and then it has a distribution over position which is some complicated function of position along the screen. It hasn't lost any definiteness in position - and indeed it is much less spread out in space than before (went from 3 extended dimensions to 2). It has only lost simplicity of representation.
How do you know which atom inside the blob is the one that hit it a moment ago? After all, they are indistinguishable. All you have left is the Hamiltonian of the blob of interacting silver atoms (and the screen, and the field) and its eigenstates, which might not even correspond to single atoms, but instead to some lattice states.
What? The note I quoted was in the approximation of 'atoms have identity'.
If you're going that far from the first, then my point about reduction of dimensions still applies.
Plus, these energy eigenstates are each for a well-defined number of atoms.
Sorry, I don't follow...
You said, Once an atom interacts with the screen, it no longer has a definite spin or even position.
In no event did any atom become LESS constrained in position by hitting the screen, and this applies whether or not you take individual identities or not. That's the first two lines.
The last line points out that you seem to think that the energy eigenstates of the screen and field might correspond to single atoms - but the eigenstates for an extended object will be multiparticle states of extreme complexity - and, at least within the energy regime we're talking about, a fixed number of particles.