Kinetic energy and path curvature in bound state systems

Abstract

The effect of path curvature on kinetic energy in bound state systems has been investigated by solving the Schrödinger equation for a particle confined to an annular region of uniform width. In the one-dimensional limit of a narrow annulus the angular behavior of the particle can be described by linear motion with an effective potential. This potential is related to the radius of curvature R by Veff=−(h/2/2m)(1/4R2). Thus, for a particle confined to a narrow channel high path curvature produces an effective stabilizing influence. An interpretation of this phenomenon is offered and several examples are examined.

Polymers in Curved Boxes

“…), the polymer would localize in regions where there is curvature. One explicit example of this is the numerical solution of Schrödinger's equation in oval shaped rings [14].…”

Bent waveguides for matter-waves: supersymmetric potentials and reflectionless geometries

“…In addition, the acceleration of the guided matter-waves towards the region of high-curvature is prevented. To explore in detail this possibility, we consider an elliptical trap [20,29,45], associated with the curve r(u) = (a cos u, b sin u), with a ≥ b > 0, and circumference L. The CIP in an elliptical trap reads The eccentricity of an ellipse is defined by ε = [1−(b/a) 2 ] 1 2 ∈ [0, 1] and can be used to quantify the deformation from a circle (for which a = b, ε = 0). For a ring of radius a = b (γ, with ε = 0), the curvature is κ(q 1 ) = 1/a and the CIP becomes constant, and the ground state density profile is uniform along the arc length q 1 .…”

Green's function for aromatic molecules in free-electron approximation

“…s function for the described model and boundary conditions(2) can be expanded into a series with orthog-= min(p,p'), p> = max(p,p')The polarisability of the ground state 10) is expressed in terms of the Green's function as follows:= (OIdjGkmn(EO + wh)dg + diGkmn(EO -wh)d1O)(5)Here E0 is the ground state energy, d1,dj are the projections of dipole momentum. As a result we have got the frequence dependencies of the polarisability under fixed model parameters: P2=2.78 A, ZO =1.24The static limit of benzene polarisability was derived and compared with the experimental data.3 This comparison shows that the theoretical results are in good agreement with the experiment for acceptable parameters values, except the inner radius p'.…”