Noncentral Forces in Three Dimensions: A Brief Survey

R. S. Kaushal

I. Lukáčs, Ya.A. Smorodinskii

Laurence Mittag, Michael J. Stephen

The use of complex variables and conformal mapping have long been useful tools in many branches of physics. However, these methods have not been widely applied in mechanics and elementary quantum mechanics. In this paper, it is shown that there are many useful applications of complex variables and conformal mapping in these subjects. It will be proven that central force problems in the plane have conformal duals. The inverse fifth power law force is self-dual in all dimensions. These results are extended to noncentral forces. Many of these results also apply to the quantum mechanics of a particle. For central force problems and certain noncentral force problems in the plane the Schrödinger equation preserves its form under conformal mapping and the inverse fifth power law force is again self-dual in all dimensions. The methods may also be applied in the presence of magnetic fields.

S. Brandt, Hans Dieter Dahmen

Avinash Khare, Rajat K. Bhaduri

We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.

S. Brandt, Hans Dieter Dahmen