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This then poses the question, how far ahead can a particle glance? If it knows it's momentum as much as it can (not explicity taking time on just yet) then it would only be able to sample a space $\frac{\hbar}{2\Delta p}$ in front or behind. What are the implications of this?  Appears to be extendable to calculus like questions, instead of infantesimal linear interpolation, use a juggling of areas proceeding in each direction.  Infinite Potential Well:  Step by step example. We are at the leftmost wall, backward is only infinity, forward is zero. This means any particles here must go forward, the density will be zero. Taking a few steps forward, ...tbc