Transition From the Infinite to the Finite Well

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We begin by looking at the ground state of a very deep, but not quite infinite, square well.  Here the potential energy function is zero everywhere except between −0.5 < x < 0.5 where it is V(x) = −|V0| which is controlled by the slider.  In addition, ħ = 2m = 1.  Notice that this energy eigenfunction, here shown in position-space, resembles that of the infinite square well in shape.  Drag the slider from left to right and see what happens.  You should notice that as |V0| gets smaller, the wave function now leaks into the two classically-forbidden regions.

Now consider a shallow well by selecting the link Show the shallow well instead.  In this animation you may again drag the slider to change |V0|, but you are now not limited to the ground-state wave function.  To see the other bound states, simply click-drag in the energy level diagram on the left to select a level.  The selected level will turn red.  Again drag the slider from left to right and see what happens. You should notice that as |V0| gets smaller, the wave function now leaks even more into the classically-forbidden regions than before.  You should also note that the number of bound states has been reduced as |V0| gets smaller.  Also note that the higher n states have a larger amount of leakage than the smaller n states for the same well depth.