Understanding quantum wires with circular bends

Lent’ and also 301s and Macucci* have carried out numerical calculations for transmission and bound states of a quantum wire with a circular bend. We shall show here that their results can be understood in detail, and very simply,. in the approximation introduced by Jensen and Koppe3 that the width of the wire is small compared to its radius of curvature. As we shall see, the results are excellent even when this condition is seriously violated. Let the circular bend have inner radius R and width G?, and subtend an angle 20 from the center of curvature. The bend is connected to entrance/exit leads which are straight, and of the same width d, as sketched in the inset to Fig. 1. In the straight section we use Cartesian coordinates (x,r), and write the Schriidinger equation