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In quantum mechanics, periodic potentials are associated with crystal lattices; natural crystals are so large that boundary effects can be ignored, which is not to say that the stydy of crystal surfaces is not important nor that the explicit consideration of boundaries and interfaces is not important. Both possibilities can be given their due recognition in the appropriate places. So considering cyclic or infinite systems is a good place to begin.

Mathematically speaking, the principal difference between the Schrödinger equation for nonrelativistic quantum mechanics and the Dirac equation for relativistic quantum mechanics is that large Schrödinger potentials act like shears, while large Dirac potentials acr like rotations. This distinction refers to the action of the coefficient matrix in the system of ordinary differential equations which remains after separating variables in many dimensions or the form which they already have in one dimension.

Large potentials have become involved for historical reasons -- symbolic solutions were more tractable the Kronig-Penney model, especially in an era when numerical computation was hard to come by. The Kronig-Penney model gets this simplification from concentrating large potentials in small regions so as to have plane waves elsewhere, both of which are readily soluble.

All seems to have gone well for nonrelativistic applications, but within recent time attempts have been made to get relativistic extensions. Several known aspects of the Dirac equation, such as the zitterbewegung and the Klein paradox, show up in the context of periodic potentials as well. Zitterbewegung and the Foldy-Wouthuysen transformation refer to the construction of positive energy wave packets, which are not of themselves eigenfunctions of the Dirac equation.

The Klein paradox refers to the fact that one and the same eigenfunction can show both positive and negative energy behavior, according to variations in the relationships between mass, kinetic and potential energy from one region to another. In particular, a delta-function potential contains such a reversal of behavior within itself, rotating wave functions in phase space rather than shearing them. In practical terms, they just shift phases, neither binding nor reflecting particles without the collaboration of still further factors.

As the dispersion relations which we have studied show, there are three distinct regions. The first is for energies well above or well below the potentials, including any mass present. In this region the wave function is that of a nearly free particle, with some impediments to propagation resulting from constructive interference of back reflections off potential peaks.

The second region, which is also similar to one of the two classical regions, features exponential behavior of wave functions with the consequent existence of bound states, or greatly suppressed tunneling between nearly bound states. The indicator of this region is the hyperbolic nature of the angle, due to a purely imaginary wave number, for long intervals, interspersed with an actual wave number in shorter intervals.

The new feature, closely related to the Klein paradox, consists of regions in which the particle energy is lower than the potential energy with its mass seell, in some intervals, yet larger than the potential, still including the mass shell, in others. When the minimum of the high shell is greater than the maximum of the low shell, the wave number is never imaginary, but regularly alternates sign. This region is evident in all the examples which have been shown. It is where the would-be delta-spike rotates rather than shears or attenuates.

next up previous contents
Next: Bibliography Up: Periodic Potentials in One Previous: Further variants