Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1994-06-14
Physics
High Energy Physics
High Energy Physics - Theory
12 pages, including 7 figures available from the authors. Los Alamos preprint LA-UR-94-1949. Invited talk at 2nd International
Scientific paper
For zero energy, $E=0$, we derive exact, classical and quantum solutions for {\em all} power-law oscillators with potentials $V(r)=-\gamma/r^\nu$, $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the classical orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, with $\mu=\nu/2-1 \ne 0$. For $\nu>2$, 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 $\nu>2$ 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.
Daboul Jamil
Nieto Michael Martin
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