Let me suppose that the infinite square well is v=infinity at x=0,L and the coresponding wave function is sqrt(2/L)*sin(2*Pi/L).

From the wave function, the probability of finding the particle on the left side (x=0~L/2) and the right side (x=L/2~L) are exactly the same.

The question is, if at x=L/2 (midpoint) the probability density is 0, how can the particle go from the left to the right?