A bouncing particle.

(a) Construct quantities having the dimensions of energy and of length when the potential energy is V(z) = mgz for z > 0 and V(z) = 1 for z < 0.

(b) An electron bounces on a horizontal surface. In its lowest energy states, the electron stays near the surface, held there by gravity. Show that the minimum thickness of the layer in which the electron is likely to be found is about 1 mm. Why is this so large?

Note: Experiments have been done on very low-energy ("ultracold") bouncing neutrons.