A classic setup (I suppose it's classic) in introductory quantum mechanics has a single photon passing through a double slitted grating. Directly across each slit of the grating is a device that can identify which particular slit a photon passed through. In this setup, the uptake is that, if you can distinguish which slit a photon passes through (i.e. localize the photon), you collapse the wavefunction and prevent any interference pattern from forming on the screen past the grating.
A practice question -- which forms the basis for this conceptual question (see below) -- localizes a photon but, this time, has it sent through an interferomter. It asks what would happen to the probability of such a particle if we also moved the right arm slightly outwards (please read the question too!). From what I understand, however, localization shouldn't bear any single consequence in interferometry. This is what I would argue:
a) Despite the fact that a single photon enters an interferometer, the photon's path can be thought to be a superposition of all possible ones. Thus, it is possible to think of the single photon entering the setup as if two photons entered different arms of an interferometer.
b) If we localize the particle, we confine its wavefunction to a certain range.
c) Consequently, unable to take on its original, full range (given by $\psi$), the photon doesn't superimpose to produce all possible intereference constructions of amplitude (the phase angle, $\phi$, has a limited set of values).
d) This could mean anything! Without knowledge of where $\psi$ is limited, many possible, albeit limited, wavefunctions result.