Transition from the Finite Well to Multiple Wells

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Previously we considered a single finite well.  What happens when there are two or more finite wells side by side?  Such a situation is called a finite lattice of square wells.  This finite lattice is modeled by a set of N finite square wells (V0 < 0) each of width b = 2a and a distance D apart from each other.  In addition, for a finite lattice, the boundary condition on the wave function is such that it is zero at the edges of the lattice, ψedges = 0.

In the animation (ħ = 2m = 1), you can change the number of wells in the lattice from 1 to 3 to 5, while maintaining the individual well's width and depth.  To see the other bound states, simply click-drag in the energy level diagram on the left to select a level.  Notice what happens to the energy level diagram.  For these particular wells, there are just two bound states possible.  What happens when we increase the lattice to include three finite wells?  Five finite wells?  What you should notice is that the number of bound states increases as the number of wells increases.  There are still two groups of states, but now each group has N individual states, where N is the number of finite wells.  Therefore with three wells there are 6 bound states (three and three) and for five wells there are 10 bound states (five and five).  As the number of wells increases, the number of bound states, therefore, will also increase.  As the number of wells approaches the number in a metal, on the order of 108, the individual states form a continuous band of states, while the energies between these bands are called gaps.