The two-body quantum mechanical problem on spheres

Abstract

The quantum mechanical two-body problem with a central interaction on the sphere S n is considered. Using recent results in representation theory an ordinary differential equation for some energy levels is found. For several interactive potentials these energy levels are calculated in explicit form.

Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton

“…For the history of this potential, see references in [10,28]. A recent book is devoted to the study of the Euclidean Kepler problem [29].…”

“…We recall that it suffices to enforce the Euclidean relation J = x ẏ −y ẋ with ẋ, ẏ taken from (1) to get the orbit equation; this was the path followed by Hamilton, who arrived at these constants as a consequence of the circle character of the hodograph, which he established directly from the Newton's equations and the law of areas. Back to the general CK case, the CK momenta P 1 , P 2 are defined for any curvature and signature type of the configuration space, and if in (28) we replace the functions W by their expressions, we obtain the parametric equations of the hodograph, with the polar angle as a parameter:…”