This article tells me that the amplitude for a photon leaving a half mirror in each of the two directions is 1 and i (for straight and taking a turn, respectively) for an amplitude of 1 of a photon reaching the half-mirror. This must be a simplification, otherwise two half mirrors in a line would result in amplitude of i photon turning at the first mirror, an amplitude of i photon turning at the second mirror, and an amplitude of 1 of photon passing through both. This means that the squared-modulus ratio is 1:1:1 and all events are equally likely, and hence the existence of the second (possibly very distant) half mirror reduces the amount of light leaving the first half-mirror to 1⁄3 from 1⁄2 the intensity. I would be shocked to find that such a result is reality since it would, among other things, allow transmission of information faster than the speed of light.
Okay, so the obvious fix is to say that Eliezer simplified things and the real rule is that there is a factor of 1/sqrt(2) to each factor. Then the squared modulus ratio of the above example is 1/2:1/4:1/4 as expected.
But then I run into my second problem: suppose that there is a photon headed at a half-mirror. Turning at the half mirror leads to a detector. Going straight leads to a set of four mirrors which brings the photon back to the starting point. This introduces a loop in the system. What is the amplitude of the light reaching the detector? Intuitively, I would expect this to be 1 or possibly less than 1. Assuming that my above factor of 1/sqrt(2) is correct, then we get an infinite sum 1/sqrt(2) + 1/sqrt(4) + … which converges to 1 + sqrt(2). This seems very wrong—we would need a factor of 1⁄2 to converge to 1, but then the previous situation gives a square modulus ratio of 1/4:1/16:1/16 or 4:1:1 which is again unexpected.
So is there a factor on each term of the half-mirror and if so what is it? Since no factor would agree with both of these setups, what have I done wrong?
What dbaupp said. But in particular you square first and then add because arriving at a different time makes the possibilities distinguishable, and so there is no interference (you don’t add the complex amplitudes).
Question regarding the quantum physics sequence:
This article tells me that the amplitude for a photon leaving a half mirror in each of the two directions is 1 and i (for straight and taking a turn, respectively) for an amplitude of 1 of a photon reaching the half-mirror. This must be a simplification, otherwise two half mirrors in a line would result in amplitude of i photon turning at the first mirror, an amplitude of i photon turning at the second mirror, and an amplitude of 1 of photon passing through both. This means that the squared-modulus ratio is 1:1:1 and all events are equally likely, and hence the existence of the second (possibly very distant) half mirror reduces the amount of light leaving the first half-mirror to 1⁄3 from 1⁄2 the intensity. I would be shocked to find that such a result is reality since it would, among other things, allow transmission of information faster than the speed of light.
Okay, so the obvious fix is to say that Eliezer simplified things and the real rule is that there is a factor of 1/sqrt(2) to each factor. Then the squared modulus ratio of the above example is 1/2:1/4:1/4 as expected.
But then I run into my second problem: suppose that there is a photon headed at a half-mirror. Turning at the half mirror leads to a detector. Going straight leads to a set of four mirrors which brings the photon back to the starting point. This introduces a loop in the system. What is the amplitude of the light reaching the detector? Intuitively, I would expect this to be 1 or possibly less than 1. Assuming that my above factor of 1/sqrt(2) is correct, then we get an infinite sum 1/sqrt(2) + 1/sqrt(4) + … which converges to 1 + sqrt(2). This seems very wrong—we would need a factor of 1⁄2 to converge to 1, but then the previous situation gives a square modulus ratio of 1/4:1/16:1/16 or 4:1:1 which is again unexpected.
So is there a factor on each term of the half-mirror and if so what is it? Since no factor would agree with both of these setups, what have I done wrong?
What dbaupp said. But in particular you square first and then add because arriving at a different time makes the possibilities distinguishable, and so there is no interference (you don’t add the complex amplitudes).
Ah good. This is a good explanation and I had been wondering how the different timing would affect it. Thanks to you and dbaupp.
To get the ratio, one needs to add the squared moduli, so 1/2+1/4+..., and that gives 1.