Two-dimensional noncentral potentials are derived in the exact classical limit of quantum mechanics where the quantum correction to the classical Hamilton-Jacobi equation is canceled. It is then shown explicitly that there exist Ermakov-type invariants I for a large class of such potentials. Some of the invariants are found to be positive definite, $I>0,$ and for the other, we have found $I=0.$ Two-dimensional noncentral potentials are derived in the exact classical limit of quantum mechanics where the quantum correction to the classical Hamilton-Jacobi equation is canceled. It is then shown explicitly that there exist Ermakov-type invariants I for a large class of such potentials. Some of the invariants are found to be positive definite, $I>0,$ and for the other, we have found $I=0.$

Physical Review AAug 6, 20039 citations

Exactly solvable noncentral potentials in two and three dimensions

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.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.

arXiv:hep-th/9310104American Journal of PhysicsNov 1, 199457 citations

SINGULAR POTENTIALS IN QUANTUM MECHANICS

Frederick L. Scarf

Jun 1, 19561 citation

I>0,

and for the other, we have found

I=0.