Alright, I’ll take a crack at this. I haven’t read the comments, so likely (I hope?) there’s a lot of duplicate information here.
1: When does the atom align itself?
I’m not 100% sure what you mean by this, so let me know if I misinterpreted the question. Consider a single electron. Think of the wavefunction as the product of the spin wavefunction (which is representable as some linear combination of spin up and spin down), and the position-space wavefunction (which is probably a pretty tight gaussian wavepacket). This goes propagating along happily until it reaches the SG aperatus. Up until now, the spin wavefunction didn’t affect the time evolution of the position wavefunction at all: this is no longer true.
Quantum mechanics is linear: If you calculate what happens to one half of the wavefunction, and what happens to the other half, and add them together, you get the whole thing. So, you start with:
A and B are some complex coefficients whose squares add to one. Now, the first term is pure spin up. As such, it will move up, while the second term will move down. This means your new wavefunction will be:
Now we get into interpretational differences. I believe in the Copenhagen interpretation, it goes something like this:
The atom hits the screen: this is a position measurement. The wavefunction collapses, and one of the position eigenstates is chosen. It’s either going to be an eigenstate of a position in the top section of the screen, or of a position in the bottom section of the screen. Now, there are multiple eigenstates for the same position: spin up or spin down, because positon measurements don’t care about spin. However, if you wind up with a top section position eignenstate, it must be spin up, as your pre-measurent wavefunction didn’t have any component that was both in the top section and spin down. Likewise, if you measure the position of the atom as in the bottom section, you must be spin down.
Now on to MWI. We need to describe the state of the screen now, so I’m going to add another term to the wavefunction. Before the silver atom hits the screen:
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Now, similarly to how the position of the atom got entangled with the spin of the particle, the state of the screen is going to become entangled with both of them, resulting in:
Now here comes the “world splitting” bit. When you look at the screen, or otherwise become causally entangled with the state of the screen in any way (that is to say, when your wavefunction depends on the state of the screen in some way).
Before this happens, you have:
Now, remember that QM is linear. As such, you can treat each term completely separately. The first term looks like a world where the screen is marked near the top, and the atom is purely spin up. The second term looks like a world where the screen is marked near the top, and the atom is purely spin down. As soon as you become entangled with the state of the screen, the spin no longer seems to be in superposition at all, but is simply up or down, depending on if we’re discussing the experiences of You_A or You_B. I would say that the world splits when you interact with the screen.
I may continue, but this level of detail is excruciating and I’m a bit burned out from it atm.
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.
Note that once an atom interacts with the screen, it no longer has a definite spin or even position.
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.
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.
Alright, I’ll take a crack at this. I haven’t read the comments, so likely (I hope?) there’s a lot of duplicate information here.
1: When does the atom align itself?
I’m not 100% sure what you mean by this, so let me know if I misinterpreted the question. Consider a single electron. Think of the wavefunction as the product of the spin wavefunction (which is representable as some linear combination of spin up and spin down), and the position-space wavefunction (which is probably a pretty tight gaussian wavepacket). This goes propagating along happily until it reaches the SG aperatus. Up until now, the spin wavefunction didn’t affect the time evolution of the position wavefunction at all: this is no longer true.
Quantum mechanics is linear: If you calculate what happens to one half of the wavefunction, and what happens to the other half, and add them together, you get the whole thing. So, you start with:
A and B are some complex coefficients whose squares add to one. Now, the first term is pure spin up. As such, it will move up, while the second term will move down. This means your new wavefunction will be:
Aleft|text{packetmovingforwardandup}right>left|uparrowright> Bleft|text{packetmovingforwardanddown}right>left|downarrowright>
Now we get into interpretational differences. I believe in the Copenhagen interpretation, it goes something like this:
The atom hits the screen: this is a position measurement. The wavefunction collapses, and one of the position eigenstates is chosen. It’s either going to be an eigenstate of a position in the top section of the screen, or of a position in the bottom section of the screen. Now, there are multiple eigenstates for the same position: spin up or spin down, because positon measurements don’t care about spin. However, if you wind up with a top section position eignenstate, it must be spin up, as your pre-measurent wavefunction didn’t have any component that was both in the top section and spin down. Likewise, if you measure the position of the atom as in the bottom section, you must be spin down.
Now on to MWI. We need to describe the state of the screen now, so I’m going to add another term to the wavefunction. Before the silver atom hits the screen:
Now, similarly to how the position of the atom got entangled with the spin of the particle, the state of the screen is going to become entangled with both of them, resulting in:
Aleft|text{packetmovingforwardandup}right>left|uparrowright>left|text{screenwithtopmark}right> Bleft|text{packetmovingforwardanddown}right>left|downarrowright>left|text{screenwithbottommark}right>
Now here comes the “world splitting” bit. When you look at the screen, or otherwise become causally entangled with the state of the screen in any way (that is to say, when your wavefunction depends on the state of the screen in some way). Before this happens, you have:
And afterwards, you have:
Aleft|text{packetmovingforwardandup}right>left|uparrowright>left|text{screenwithtopmark}right>left|text{You}_Aright> Bleft|text{packetmovingforwardanddown}right>left|downarrowright>left|text{screenwithbottommark}right>left|text{You}_Bright>
Now, remember that QM is linear. As such, you can treat each term completely separately. The first term looks like a world where the screen is marked near the top, and the atom is purely spin up. The second term looks like a world where the screen is marked near the top, and the atom is purely spin down. As soon as you become entangled with the state of the screen, the spin no longer seems to be in superposition at all, but is simply up or down, depending on if we’re discussing the experiences of You_A or You_B. I would say that the world splits when you interact with the screen.
I may continue, but this level of detail is excruciating and I’m a bit burned out from it atm.
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.