Abstract
We solve a Schrödinger equation with a potential having two periods, whose ratio is arbitrary. This is one of very few cases for which the solution can be fully discussed. If is rational, i.e., commensurate, the eigenfunctions are Bloch states, and the energy levels fall into a spectrum of continuous bands. If is a typical irrational number, the eigenfunctions are localized with exponentially decaying tails, and each has a distinct center, just as in a random system. The spectrum (called pure point) covers all energies, but only a finite number of energies belong to wave functions appreciable in a given region. A third rarely encountered, but currently interesting, type of spectrum, the singular continuous, occurs when is a "Liouville number," a special irrational number "infinitely close" to rational numbers. This case is also concretely illustrated and interpolates between the other two possibilities. The time evolution of wave packets is also discussed.
- Received 2 January 1984
DOI:https://doi.org/10.1103/PhysRevB.29.6500
©1984 American Physical Society