In Feynman's NZ lectures (and consequent book) “QED – The Strange Theory of Light and Matter”, he gives a model for optics.
He describes a probability amplitude for a photon to be detected after being emitted from a source. The amplitude is a complex number, whose angle rotates at a constant rate (depending on the photon frequency), and whose modulus is proportional to $1/l$, where $l$ is the path length. The total amplitude is the sum of amplitudes from different paths. The probability is the total amplitude's square-length. This is a simplified model for the Path Integral.
I have build a Mathematica simulation for this method. I tried to simulate a single-slit experiment: a source (at the origin), a slit (at x-position $d$, y-positions $-yrange \to yrange$), and a detector at varying positions $(1,h)$. For each detector, I run over different paths (like the blue and yellow paths below). Each path is two straight-lines: origin to some middle point $(d,y)$, and from the slit to the detector. I sum over all paths with $y$ as a parameter. The photon wavenumber is $k$. The probability is not normalized in this method.
For $k=20$:
As you can see, I do not get a $Sinc^2$. What am I missing?
Best Answer
It seems that the slit was too close to the source and to the screen. I have also made the wavenumber higher.
When both distances are $50$, and $k=1,000$:
For two slits (source to slits $=500$, slits to screen $=500$, $k=500$, slit-width $=0.2$, slit distance $=1$):