On the one-dimensional approximations to the quantum mechanics of π -electrons in chain and ring systems

Abstract

A critical examination is made of the one-dimensional approximations for the quantum mechanical motion of almost-free electrons confined near a general space curve. A special set of coordinates which separate the longitudinal from the transverse directions for the general curve are introduced to specify the motion. The Laplace-Beltrami operator is found and Schrödinger’s equation is investigated when the transverse motion is limited to a small displacement. The special case where the curve has isolated sharp kinks is shown to be soluble in terms of certain boundary conditions at the points of large rate of change of curvature. When the curvature varies slowly the transition energies are shown to follow from the onedimensional longitudinal approximation to a certain order in the transverse displacements. However, the absolute energies are strongly dependent on the transverse motions.