Occasionally people direct me to the wiki Less Wrong about various things. Here's a critique of the explanation of quantum mechanics. Configuations and Amplitude.
Eliezer_Yudkowsky describes a classic Mach-Zehnder interferometer and some complex number maths to describe the amplitude flows. He then asserts that passing through a silvered mirror multiplies the amplitude by 1 (i.e. no change) and being reflected multiplies the amplitude by i.
To extend this I've included a modified Mach-Zehnder with two extra mirrors at B, which has the net effect of multiplying by i twice - or -1 for the amplitude flow. Keeping the path length the same I've added four mirrors at C which multiplies by i four times - i x i x i x i = 1, and therefore makes no change.
Let's work out the results.
Original, D-E = (1+0i)i + (0-i) = i + 0 - i = 0 D-F = (1+0i) + (0-i)i = 1 + 0i + 0i - -1 = 2 Modified D-E = (-1+0i)i + (0-i) = -i + 0 -i = -2i D-F = (-1+0i) + (0-i)i = -1 + 0i + 0i - -1 = 0
So in our modified setup we see all the photons arriving at E.
Of course we could work all this out from a proper physics text book. This tells us that reflection or refraction at the front of a mirror incurs a phase shift of half a wavelength, at the back of a half silvered mirror incurs no phase shift. If you add two waves together of opposite phase the cancel, if the same phase the add together. Two phase shifts shifts back to the starting phase.
Working this out
Original A-D via B, two phase shift (front of mirror A and front of mirror B) A-D via C, one phase shift (front of mirror C) so A-E via B, two phase shifts (the back of mirror D doesn't cause another one) A-E via C, one phase shift (passing through D doesn't cause another one) therefore opposite phase and they cancel. A-F via B, two phase shifts (passing through D doesn't cause another one) A-F via C, two phase shifts (reflecting from D causes an extra one) therefore same phase and they add. Modified, A-D via B, four phase shifts A-D via C, five phase shifts therefore A-E via B, four phase shifts A-E via C, five phase shifts therefore opposite phase and they cancel A-F via B, four phase shifts A-F via C, six phase shifts therefore same phase and they add.
This is helpful, we've two competing theories that predict different experimental results. Therefore we can carry out an experiment and tell the two theories apart and find out which one is right. Sadly I'm not in a position to do this but my money is on the physicists getting it right.