Adel,
you have to choose uncountable real random numbers uniformly. Every real number has the probability zero to choose.
But you are right, it sounds impossible to do.
Now to my further questions:
There are gaps in the explaination. So, I tried to fill these gaps by thinking about. But your answer showed me, I was wrong.
My main problem is on page 3, the red part. Up to this place everything is clear to me. But how did you get the Schrödinger equation and more importantly what is the wave function. Before you spoke about random lines etc. (and I assumed you have a probability distribution for these random lines, then the dynamics is given by a Fokker-Planck equation etc. etc.)
Interestingly, your simulation results (Fig 3, 4 and 5) support my assumption: you simulate the probability distribution of a Fokker-Planck equation (with constraints, i.e. you put it in the box). This Fokker-Planck equation has the same ground state then the Schrödinger equation (but a probability distribution has to be positive everywhere).
I wrote my PhD thesis about this connection (using it in the evolutionary algorithms). The correct name is Fisher-Eigen equation (a reaction diffusion equation)
Show me where I'm stupid to follow you.
Best
Torsten