Exact, E = 0, classical and quantum solutions for general power-law oscillators
- Los Alamos National Lab., NM (United States)
- Ben Gurion Univ. of the Negev, Beer Sheva (Israel)
For zero energy, E = 0, we derive exact, classical and quantum solutions for all power-law oscillators with potentials V(r) = {minus}{gamma}/r{sup {nu}}, {gamma} > 0 and {minus}{infinity} < {nu} < {infinity}. When the angular momentum is non-zero, these solutions lead to the classical orbits {rho}(t) = [cos {mu}({var_phi}(t) {minus} {var_phi}{sub 0}(t))]{sup 1/{mu}}, with {mu} = {nu}/2 {minus} 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. Also, there are normalizable discrete, yet unbound, state which correspond to unbound classical particles which reach infinity in a finite time. These and other interesting comparisons to the classical system will be discussed.
- Research Organization:
- Los Alamos National Lab., NM (United States)
- Sponsoring Organization:
- USDOE, Washington, DC (United States)
- DOE Contract Number:
- W-7405-ENG-36
- OSTI ID:
- 10163076
- Report Number(s):
- LA-UR-94-1949; CONF-9403139-1; ON: DE94014436
- Resource Relation:
- Conference: Workshop on harmonic oscillators,Cocoyoc (Mexico),23-25 Mar 1994; Other Information: PBD: [1994]
- Country of Publication:
- United States
- Language:
- English
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