Shannon entropy of a particle on a conical surface

Abstract

In this work we solve the time-independent Schrödinger equation of a particle restricted to move on the surface of a circular cone of finite height. The energy eigenvalues, as well as the corresponding wave functions, are obtained analytically as a function of, r and ϕ, the radial distance to the apex, 0≤r≤r_0, and the angular variable around the axis of the cone. We compute the Shannon entropy of this system in both configuration and momentum space as a function of r_0 and θ_0, the angular semi-aperture of the cone. In configuration space, the Shannon entropy decreases, signaling a more pronounced localization, as either r_0 or θ_0 diminish; in momentum space, an opposite behavior happens, i.e., the Shannon entropy increases when either, r_0 or θ_0, decrease. We also compute the radial standard deviation; we find that the Shannon entropy better describes the localization-delocalization phenomena. The present results agree with those previously published for a particle confined to a circle of radius r_0, which corresponds to θ_0=π⁄2 in the present case.