Wave function confinement via transfer matrix methods

Abstract

A new iterative method for calculating energy levels and wave functionsThe exact transfer matrix approach used in studying sectionally constant potentials in one dimension is generalized to cylindrical and spherical geometries, where the potential depends only on radius. In each geometry two transfer matrices suffice to completely describe the wave function: one for handling a discontinuity in potential and one for handling a delta-function potential barrier. This method is then applied to the problem of confining a wave function in a cylindrical configuration using only a series of carefully placed delta function potential barriers. It is found that confinement can be made to increase nearly exponentially with the number of barriers if placed correctly, but that this arrangement has an exponentially sharp dependence on both barrier position and energy.

Localization and the invariant probability measure for photonic band gap structures

“…(9), after which invoking the assumption of a uniform pdf for the invariant measure will produce a compact formula. Without loss of generality 18 we can assume that the unit vector ( ) θ z in Eq. (9) can be expressed as:…”

“…(5) be labeled n z which will be, for purposes of this algorithm, a unit vector of the that after multiplication by the th n transfer matrix18 we have the vector:…”