12 - Partially and Completely Periodic Potentials

Summary

A problem of considerable interest is the theory of motion of electrons in a bulk or in a nano sized crystalline solid. Whereas the potential in a bulk solid may be considered to be periodic in entire space, that in a nano sized solid is periodic in a finite region of space. In this chapter, we develop the formalism to study the motion of a particle in a one-dimensional potential which is periodic in a finite part of the space, and the one which is periodic in entire space. In the limit of the potential extending from −1to1, the results for the potentials periodic in the finite part of space reproduce those for the potentials periodic in entire space. We will see that completely periodic potentials can also be treated invoking symmetry considerations.

Potential Periodic in a Finite Region of Space

Consider a potential V(x) which is periodic with period a in the finite region of space 0 ≤ x ≤ Na such that

whereas on the remaining part of the x-axis,

The solution of the Schrödinger equation corresponding to the potential defined in (12.1) and (12.2) is obtained by solving it in different regions and matching the solutions at the boundaries of the adjoining regions.

We find first the solution in the region 0 ≤ x ≤ Na in which the potential is periodic. Consider the interval na ≤ x ≤ (n + 1)a which we call the nth interval. Since, due to (12.1), the potential in nth interval is same as that in (0, a), the linearly independent solutions of the Schrödinger equation in nth interval are same as those in (0, a).