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In classical mechanics a billiard ball bounces back when it hits the side
of the billiard. In the quantum world it might actually "tunnel"
through. Let me make this a little clearer. Classically a particle moving
in the following potential would just be bouncing back and fourth between
the walls. This can be easily seen from conservation of energy: The
kinetic energy can not go negative, and the total energy is conserved.
Where the potential is larger than the total energy, the particle cannot
go. In quantum mechanics this is different, and particles can penetrate
these classically forbidden regions, escaping from their cage.