I am trying to see if in Double slit experiment does the results change if you replace the screen behind. My thought experiment is as follows.
1) set up Double slit experiment with possibility of changing the screen where photons land
2) Fire one photon at a time
3) After each photon is detected, replace the screen with exact replica of the screen
4) After sufficient photons are fired (and detected on separate screens) superimpose the positions of all the photons
Question is - Will the superimposed positions show interference pattern?
One can also try changing screens after a group of photons are detected on a screen
Since the interference pattern can be detected with a single photon-the result is
||a|1>+b|2>||^2 and it's the fact that expanding this out gives terms |1>|2> and |2>|1>, that describe the interference-the superposition of N independent photons is sum_i || a_i |1>+b_i|2>||^2 and the interference term is, as usual, the mixed term, the coefficients of the terms |1>|2> and |2>|1>; even though the individual photons don't interfere with each other, since they're assumed independent.
The a_i and the b_i aren't all identical, they're random, complex, numbers; they just satisfy |a_i|^2+|b_i|^2=1, independently for any value of i.
For an N-photon state, the intensity would be
|| sum_i ( a_i|1>+b_i|2>)||^2
It suffices to expand out and compare.
If the N tries were done with an M-photon state, then one would have, of course
sum_j ||sum_i (a_ij|1>+b_ij|2>)||^2
with i=1,2,..,N and j=1,2,..M.
Taylor's experiment is described here: http://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/old_literature/Taylor1909.pdf
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