## Transition from the Finite Well to Multiple Wells

*Please wait for the animation to completely load.*

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 (*V*_{0 }< 0) each of width *b* = 2*a* 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, ψ_{edges }= 0.

In the animation (*ħ* = 2*m* = 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
10^{8}, the individual states form a continuous* band* of
states, while the energies between these bands are called *gaps*.