Pick a line of size “l” throw a random number denoting a position (“p”) on the line and associate a line whose length (“li”) is also chosen randomly but cannot exceed the length “l”. Set a constraint with a particular relation between “l”, ”p”, and “li” such that if this relation holds ignore the outcome, otherwise register the position. Boom, the outcome is the solution for Schrödinger equation for a particle in an infinite potential well (probability wave sin^2) (Fig. 1). My line of thought was: if nature is made of math then the best place to start is with a line. And what could be happening on this line. Nothing much, really, a point and a piece of line over and over. For the next energy level divide the line using natural numbers (how convenient). Simple line and simple rules lead to this gigantic dance of reality. It has been a dream of many that we owe our existence to some kind of automata. Well, this is it, maybe.
The program is listed here It is written in liberty basic and you can download it here. And if that is not enough I simulated two particles where the second acts like a potential. The two particles interact according to some basic rule (see the program). When you make the potential narrow i.e. the energy high, the first particle gets constrained and its probability goes to zero at the potential (fig. 2). When the second particle has a wide base then it very much acts like a square well and the probability looks exponential inside it (fig. 3). And tunneling phenomenon is clear, continuity automatically satisfied. In other words Schrödinger equation popped out of some geometry with rules. Crazy enough? No? Here is some more: I can make the particles interact without their wave functions overlapping just by not restricting the size of the associated length “li”. That is spooky action at distance. Left as an exercise.