Exact, E=0, Classical and Quantum Solutions for General Power-Law Oscillators

Abstract

For zero energy, $E=0$, we derive exact, classical and quantum solutions for {\em all} power-law oscillators with potentials $V(r)=-γ/r^ν$, $γ>0$ and $-\infty <ν<\infty$. When the angular momentum is non-zero, these solutions lead to the classical orbits $\r(t)= [\cos μ(þ(t)-þ_0(t))]^{1/μ}$, with $μ=ν/2-1 \ne 0$. For $ν>2$, the orbits are bound and go through the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. The unbound orbits are also discussed in detail. Quantum mechanically, this system is also exactly solvable. We find that when $ν>2$ the solutions are normalizable (bound), as in the classical case. Further, there are normalizable discrete, yet {\it unbound}, states. They correspond to unbound classical particles which reach infinity in a finite time. Finally, the number of space dimensions of the system can determine whether or not an $E=0$ state is bound. These and other interesting comparisons to the classical system will be discussed.