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One of the very first basics of the double slit experiment is that you have a single photon source that is directed at a reference screen. There, in contrast to the physics of e.g. a pistol bullet, the photon particles are distributed completely randomly. Is this correct?

Then I have difficulties to understand this. Because if I can develop a single photon source and direct it to a target and there all photons hit somewhere, I would assume that my device has a problem and not that there is a physical component here due to which all photons distribute themselves randomly on the target.

In addition, I do not understand, in which area random distribution takes place? Somehow a target area must be defined from the photon source, otherwise the photons would be distributed on the whole screen and beyond.

So is the photon source actually a "light beam" like a laser beam, which in the double-slit experiment completely covers the slits and within this beam, however, only single photons are emitted?

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You take any source of light and create a beam of desired width using solid non-transparent obstacles. Then you put dark half-transparent absorbers in the path of the beam, e.g. darkened glass slabs. When there is enough of these obstacles in the path, the light will get so weak that there is a good argument for there being less than one quantum of energy $hf$ between the double slit and screen at any time. With a weak-enough light, the screen where the beam falls will evolve quasi-random spots where the screen interaction with the light resulted in chemical change (or electric current detection in CCD detector). The whole pattern is not completely random, because it still manifests interference stripes.