Classical and quantum dynamics in an inverse square potential

Abstract

The classical motion of a particle in a 3D inverse square potential with negative energy, E, is shown to be geodesic, i.e., equivalent to the particle's free motion on a non-compact phase space manifold irrespective of the sign of the coupling constant. We thus establish that all its classical orbits with E < 0 are unbounded. To analyse the corresponding quantum problem, the Schrödinger equation is solved in momentum space. No discrete energy levels exist in the unrenormalized case and the system shows a complete “fall-to-the-center” with an energy spectrum unbounded by below. Such behavior corresponds to the non-existence of bound classical orbits. The symmetry of the problem is SO(3) × SO(2, 1) corroborating previously obtained results.

Motion of Massive Particles in Rindler Space and the Problem of Fall at the Centre

“…Therefore in actual scenario this is not the potential hill but is potential well. But unlike the conventional scenario of fall at the centre problem, there is no singularity at the origin (x = 0), or at the centre of the gravitating object [14,15,16]. The value of V (u) is −0.5Cα 2 at the origin and −→ −1 for large u values.…”

“…(3) we have shown the geodesics of motion for three different u 0 values. Now to complete the study of free fall at the centre in our modified formalism, let us next investigate the quantum mechanical motion of the particle in inverse square attractive potential [14,15,16]. We consider the bound state problem, i.e., E < 0.…”