Quantum mechanics of Kepler orbits

Arno Böhm

Arno Böhm

P. Enders

‘Aequat causa effectum’ (the cause is equal to the effect) is an old metaphysical rule, which seems, however, to be violated for the single trajectories in spherically symmetric force fields, as the Kepler orbits are not spherically symmetric. The reason for the latter is the lower symmetry of the initial conditions. The rule is reestablished, if appropriate sets of trajectories (i.e, sets of initial conditions) are considered. The corresponding hodographs are also explored. The fact, that the Kepler orbits are plane, is derived by means of logical and symmetry arguments, without using angular momentum conservation. Spherically symmetric sets of Bohr orbitals are constructed by means of a sum rule for the spherically harmonic functions (Unsöld’s theorem). This results in a less known similarity between classical and quantum mechanical systems. Metaphysical rules do not solve concrete physical problems, but encourage to explore them more profoundly.

Guillermo Chacón‐Acosta, Héctor H. Hernández

Jason D. Tran, Leanne Doughty, James K. Freericks

Central force problems involve coupled linear differential equations of motion. By decoupling the differential equations, we show how to efficiently solve for the orbits of the Kepler problem without any complicated integrals. Introducing quantization provides a deep connection between the solution for these classical orbits and the quantum Hamiltonian of hydrogen for states that have definite total angular momentum.

S. D. Boris, S. Brandt, Hans Dieter Dahmen +2 more

The quantum-mechanical motion of Gaussian wave packets on elliptical and hyperbolic orbits is calculated and shown in plots. The approximation scheme makes explicit use of energy and angular-momentum conservation thus guaranteeing controlled accuracy of the approximation over long-time intervals. This allows the study of the revivals, a phenomenon of particular interest in Rydberg atoms on elliptical orbits. A comparison with the behavior of a classical phase-space distribution of the same position and momentum widths is carried out.