Exact, E=0, Solutions for General Power-Law Potentials. II. Quantum Wave Functions

TL;DR

This paper derives exact quantum solutions at zero energy for all power-law potentials, revealing conditions for normalizability, bound states, and the effects of higher dimensions on binding.

Contribution

It provides a comprehensive set of exact solutions for all power-law potentials at zero energy, including novel findings on normalizability and the impact of extra dimensions on bound states.

Findings

Normalizable solutions for  > 2 and l  1

Normalizable solutions for  <  2 are discrete but not bound

Higher dimensions ( > 4) create effective barriers enabling bound states at  > 2

Abstract

For zero energy, $E=0$, we derive exact, quantum solutions for {\it all} power-law potentials, $V(r) = -\gamma/r^{\nu}$, with $\gamma > 0$ and $-\infty < \nu < \infty$. The solutions are, in general, Bessel functions of powers of $r$. For $\nu > 2$ and $l \ge 1$ the solutions are normalizable; they correspond to states which are bound by the angular-momentum barrier. Surprisingly, the solutions for $\nu < -2$ are also normalizable, They are discrete states but do not correspond to bound states. For $2> \nu \geq -2$ the states are unnormalizable continuum states. The $\nu=2$ solutions are also unnormalizable, but are exceptional solutions. Finally, we find that by increasing the dimension of the \seq beyond 4 an effective centrifugal barrier is created, due solely to the extra dimensions, which is enough to cause binding. Thus, if $D>4$, there are $E=0$ bound states for $\nu > 2$ even for $l=0$. We discuss the physics of the above solutions are compare them to the classical solutions of the preceding paper.