pines-demon said:
Maybe you should also start to react with "of course" from time to time, otherwise I have little idea where I lost you, or where we disagree. Take for example
gentzen said:
If you define it like this, then ##S_x## and ##S_p## are already matrices (or rather operators),
Just answer something like: "Of course, my ##S_x## and ##S_p## are operators. Observables in QM are self-adjoint operators after all." Or also: "No, I disagree. My ##S_p## is exactly analogous to ##S_z## in spin measurements."
gentzen said:
basically ##S_x=\int_0^\infty dx\, |x\rangle\langle x|## and ##S_p=\int_0^\infty dp\, |p\rangle\langle p|##.
also here "Yes, basically. But my ##S_x## has eigenvalues 1 and -1, while yours has 1 and 0."
gentzen said:
If you want to be more accurate, it would be ##S_x=\int_{-\infty}^\infty dx\, sign(x)|x\rangle\langle x|##. But the crucial part is that ##|x\rangle\langle x|## got already formed.
or here: "Yes, I agree." Or if you prefer: "Why do you bother me with this. Why is it important?"
pines-demon said:
This is getting very cryptic. I take ##S_\theta=S_x\cos\theta+S_p\sin\theta=\alpha S_x+\beta S_p##, I do what then?
Looks to me like you think I am being cryptic. Correct?
If you have an observable ##S_\theta## and want to check whether it is analogous to the spin observables in Bell measurements, I guess you should check whether its eigenvalues are +1 and -1.
pines-demon said:
##S_\theta^2## or ##S_\theta S_\theta^\dagger##? Either way I get interference term right? Is not that different to your calculation. What am I missing?
No, you can compute ##S_\theta^2## or ##S_\theta S_\theta^\dagger## if you want, but it doesn't seem relevant to anything we are discussing here.
pines-demon said:
1) Please be explicit on what is the calculation to perform to check if there is interference.
2) Why is it needed?
The "Why is it needed?" is a good question. For the Bell experiment, the more relevant question would be to check the spectrum/eigenvalues of your ##S_\theta##. Even so the spectrum/eigenvalues are hard to compute, checking a specific eigenvalue is doable, "in principle".
What I mean by "interference" is terms which can have a negative sign. Say if I have light in a horizontally homogeneous layer, I will often first compute amplitudes ##A_{up}## and ##A_{down}##. If I am only interested in the general intensity distribution without standing waves, I could compute ##I_{up}+I_{down}## with ##I_{up}=|A_{up}|^2## and ##I_{down}=|A_{down}|^2##. If the light is perfectly monochromatic and I am interested in the standing waves, I could compute ##I_{up}+I_{down}+2\operatorname{Re}(A_{up}^*A_{down})##. The first two terms are non-negative, hence it is the third term which can have a negative sign. (But of course, it goes in both directions, because there is also constructive interference.)