Interactive Problem #3, bound states of spatially periodic potentials

Crystalline solids, like Cu, Al, Ag, Au, C(diamond), C(graphite), Si, Ge, GaAs, CdTe, NaCl, KCl, LiF, RbI are made of atoms arranged in a crystalline structure. In such a circumstance, when the atoms are relatively close together, the electron states usually are not associated with a particular atom, but rather, extend throughout the crystal. It turns out that one can "build" these extended (non-local) crystal states from the states of individual atoms, and that this is conceptually and computationally a very succesful and useful approach.

Suppose we make a 1-dimensional crystal from an array of delta-function atoms, that is, an infinite array of attractive delta-function potentials of identical strength and equally spaced along the x axis. Using the notes from the earlier post on electron eigenstates in spatially periodic systems, see if you can find the approximate energies of all the electron eigenstates that come from the single bound state of a single, isolated attractive delta-function potential. Perhaps you can identify a "small parameter" that arises and obtain the energies to lowest order in which the energies are not all exactly equal to the starting bound state energy. (Boldness and daring may be called for here.)

b) Is there any redundancy in your states? Define a basis that spans this subspace of states. (Thoughtfulness and careful inspection may be of value here.)

c) Find a good way to present your results regarding the energies of these states. How do they compare to zero? What are possible values of the work-function for this system? How do the energies of the lattice states compare to the energy of the atomic state; how do they depend on "atom spacing", a? and so on.