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

Abstract

For zero energy, $E=0$, we derive exact, quantum solutions for {\it all} power-law potentials, $V(r) = -γ/r^ν$, with $γ> 0$ and $-\infty < ν< \infty$. The solutions are, in general, Bessel functions of powers of $r$. For $ν> 2$ and $l \ge 1$ the solutions are normalizable; they correspond to states which are bound by the angular-momentum barrier. Surprisingly, the solutions for $ν< -2$ are also normalizable, They are discrete states but do not correspond to bound states. For $2> ν\geq -2$ the states are unnormalizable continuum states. The $ν=2$ 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 $ν> 2$ even for $l=0$. We discuss the physics of the above solutions are compare them to the classical solutions of the preceding paper.