Rohan de Silva said:
I know that photon is not a "classical" particle and it's position adhere to Schrodinger equation.
No, this is not correct. The Schrodinger equation, as
@vanhees71 mentioned, only works for non-relativistic particles, which means, roughly, particles moving much slower than the speed of light--or more precisely, particles that do not have any amplitude to move at or near the speed of light. Photons obviously do not meet this restriction.
Rohan de Silva said:
Do they close the slit when the entire Schrodinger equation probability distribution passes the slit??
They close the slit based on the calculation I described earlier, which has nothing to do with the Schrodinger equation at all. It's a calculation based on the (wrong) assumption that a photon is a classical particle that moves at the speed of light.
There is no way to close the slit only when the entire wave function is past the slit, if you want to close the slit during the experiment at all. Until the particle is observed hitting the detector at the other end, the wave function will have some nonzero amplitude to not yet have passed the slit.
Rohan de Silva said:
I have read that the double-slit experiment has been done with massive atoms as well. So, don't they obey the Schrodinger equation?
They do--at least, they do as long as the atoms are moving much slower than the speed of light (or more precisely, that whatever state the atoms are in, it has no amplitude to move at or near the speed of light). I don't know if delayed choice experiments have been done with such atoms, though. If they were, the time at which the delayed choice is made would have to be calculated based on some reasonable estimate of the speed of the particle, such as its expectation value for velocity. As I noted above, there is no way to wait until the particle's wave function is entirely past the slit before closing it, if you want to do a delayed choice experiment at all.