Re: what is the mathematical basis for stating a single photon can be split

The answer is ... experiments!

When scientists shine a bright light source through two slits, they see an interference pattern cast on a distant wall. That seems pretty easy to explain: photons which pass through one slit interfere with those which pass through another slit. Since the bright light source produces so many photons each second, there are plenty of photons passing through each slit at any time, so there will always be pairs which can interfere with each other.

Will a single photon interfere with itself? It's easy enough to find out: just decrease the brightness of the light source, so that it emits only a few photons each second. Let's say that the slits are 1 meter from a distant wall. A photon will cover that distance in (1 meter) / (300,000,000 meter/second) = 3.3 nanoseconds. If the light source produces, say, 10 photons each second, and each one takes only 0.000 000 003 seconds to fly from the slits to the wall, then each photon will be alone as it moves through the apparatus.

Physicists have been doing this sort of experiment for a long, long time. For example, back in 1909, G. I. Taylor set up some equipment in one room of his house: a gas lamp and some screens, a pair of slits, a photographic plate. He put many screens between the lamp and the slits, to decrease the apparent brightness of the lamp until fewer than 1 photon per second was reaching the slit. You can read about his experiment on one of my lecture pages , or in Proceedings of the Cambridge Philosophical Society, vol 15, p. 114 (1909), if you have a copy. He exposed the photographic plate for periods as long as three months (!), and found interference patterns.

I am supervising students as they perform a modern version of this experiment this week. They shine a very dim light through a pair of slits, and detect the arrival of each photon on a distant wall using a photomultiplier tube. If they record the position of each photon, and move the tube around from place to place on the wall, they can build up a picture which shows bright and dark fringes. From the number of photons they count per second -- just a few hundred -- they can figure out that there are big gaps between successive photons in the apparatus. Again, experiments show that even a single photon will interfere with itself as it passes through two slits.