KronigPenney

The Kronig-Penney model is a simplified quantum mechanical model used to illustrate how energy bands arise for an electron in a one-dimensional periodic potential. Proposed in 1931 by Ralph Kronig and S. Penney, it treats the crystal as a regular array of potential barriers or wells and provides an exactly solvable example of Bloch's theorem in action.

In the model, the potential V(x) is periodic with period d = a + b, consisting of a region of width a with one potential value and a barrier of width b with height V0. The Schrödinger equation is solved in each region, and Bloch's condition psi(x + d) = e^{i k d} psi(x) is applied. This yields a dispersion relation linking energy E and Bloch wave number k; the relation is usually written as cos(k d) = F(E; a, b, V0), where F depends on the details of the matching at the interfaces.

Energies for which |F(E)| ≤ 1 form allowed bands; those with |F(E)| > 1 lie in gaps where no Bloch states exist. The band structure becomes more pronounced as barrier height V0 or barrier width b increases; broader wells (larger a) and differing d shift and reshape the bands. In the weak-potential limit the model reproduces nearly-free electron behavior; in the tight-binding limit it resembles a chain of isolated sites.

Significance: The Kronig-Penney model is a foundational pedagogical tool for understanding band structure in solids, highlighting how periodic potentials generate bands and gaps. While simplified to one dimension with square-type potentials, it complements more realistic approaches such as the nearly-free electron model and the tight-binding model.