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Sniffnoy comments on Configurations and Amplitude - Less Wrong
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Can you clarify the question? Do you mean turning the mirror by a quarter-turn from its current orientation, so it's diagonal in the other direction? I compute that if you do that it should work out with 1/4 chance of E, 1/4 chance of F, and 1/2 chance of neither (if it's reflected back towards B or C it will never reach either). Exactly like the classical case, actually...
Edited to clarify - I agree re quarter turn, but it turns out he means half turn. I think our thought-experiment half-silvered mirrors are unchanged by a half turn.