Here's what I was missing: the magnitudes of the amplitudes needs to decrease when changing from one possible state to more than one. In drawing-on-2d terms, a small amount of dark pencil must change to a large amount of _lighter_ pencil, not a large amount of equally dark pencil. So here's what actually occurs (I think):

A photon is coming toward E (-1,0)

A photon is coming from E to 1 (0,-1/sqrt(2))
A photon is coming from E to A (-1/sqrt(2),0)

A photon is coming from E to 1 (0,-1/sqrt(2))
A photon is coming from A to B (0,-1/2)
A photon is coming from A to C (-1/2,0)

A photon is coming from E to 1 (0,-1/sqrt(2))
A photon is coming from B to D (1/2,0)
A photon is coming from C to D (0,-1/2)

A photon is coming from E to 1 (0,-1/sqrt(2))
A photon is coming from D to X (0,1/2sqrt(2))+(0,-1/2sqrt(2)) = (0,0)
A photon is coming from D to 2 (1/2sqrt(2),0)+(1/2sqrt(2),0) = (1/sqrt(2),0)

Detector 1 hits 1/2 of the time and detector 2 hits 1/2 of the time.

Comment author:[deleted]
15 September 2013 09:42:31AM
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0 points
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What I was about to say. It really doesn't matter yet, but it's better to get the reader used to unitarity straight away. (Though I wouldn't explicitly mention unitarity this early -- I'd just replace the rule with "Multiply by 1/sqrt(2) when the photon goes straight, and multiply by i/sqrt(2) when the photon turns at a right angle" and everything that follows from that. If the maths gets too complicated with all those denominators, just make the initial amplitude -sqrt(2) rather than -1.)

## Comments (378)

OldHere's what I was missing: the magnitudes of the amplitudes needs to decrease when changing from one possible state to more than one. In drawing-on-2d terms, a small amount of dark pencil must change to a large amount of _lighter_ pencil, not a large amount of equally dark pencil. So here's what actually occurs (I think):

A photon is coming toward E (-1,0)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from E to A (-1/sqrt(2),0)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from A to B (0,-1/2) A photon is coming from A to C (-1/2,0)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from B to D (1/2,0) A photon is coming from C to D (0,-1/2)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from D to X (0,1/2sqrt(2))+(0,-1/2sqrt(2)) = (0,0) A photon is coming from D to 2 (1/2sqrt(2),0)+(1/2sqrt(2),0) = (1/sqrt(2),0)

Detector 1 hits 1/2 of the time and detector 2 hits 1/2 of the time.

*0 points [-]What I was about to say. It really doesn't matter

yet, but it's better to get the reader used to unitarity straight away. (Though I wouldn'texplicitlymention unitarity this early -- I'd just replace the rule with "Multiply by 1/sqrt(2) when the photon goes straight, and multiply by i/sqrt(2) when the photon turns at a right angle" and everything that follows from that. If the maths gets too complicated with all those denominators, just make the initial amplitude -sqrt(2) rather than -1.)