Having read this question and answers to it, I've learned that somehow two light beams from independent sources can actually produce interference pattern, if the properties of their sources are good enough.
Now, this conflicts with my understanding of quantum particles. I suppose that interference of independent beams means interference between pairs of particles from each beam. Consider a pair of non-relativistic non-interacting bosons such that it can be described by Schrödinger equation. Its state function, up to normalization, would be $$\Psi(\vec r_1,\vec r_2,t)=\psi_1(\vec r_1,t)\psi_2(\vec r_2,t)+\psi_2(\vec r_1,t)\psi_1(\vec r_2,t).$$
Let now $\psi_1(\vec r,t)$ be a 2D gaussian wave packet going along $y=x$ axis, and $\psi_2(\vec r,t)$ be a similar wave packet going in direction of $y=-x$. Obviously there exists an area where they "intersect". Let this point be around $(x,y)=(0,0).$ At this point we might place a detector, along $y=0$, which would show us the intensity of the particle beam, which would be governed by the following formula:
Now I wasn't able to find this integral analytically, but numeric calculations show that there's no interference pattern on the screen, the intensities of both beams just add up.
This is what I get for particle density in $(x,y)$ space — evaluated as in $(1)$:
And this is what I'd expect based on the answers to the question mentioned above (this was generated as a probability density for single particle in two-packets state):
So the question: what's so special about photons that they do exhibit interference pattern, while usual non-relativistic bosons obeying Schrödinger equation don't? I suppose the core reasons might be some of:
- Non-relativistsness of Schrödinger equation, which my analysis was based on
- Different form of equations governing evolution of light, i.e. Maxwell's equations vs Schrödinger's one.
- Something related to QED, which is not taken into account in QM
- My mistake
- Something other
What are the real reasons for this discrepancy?