Zero-Energy Quantum States for a Class of Noncentral Potentials and an Exact Classical Limit

R. S. Kaushal

R. S. Kaushal

R. S. Kaushal

Adam J. Makowski

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 $I=0.$

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.

R. S. Kaushal

With a view to obtaining an exact analytic solution to the time-dependent Schr\"odinger wave equation for noncentral harmonic and anharmonic potentials in two dimensions, we make use of an ansatz for the eigenfunction in conjunction with the methods of Burgan et al. [Phys. Lett. 74A, 11 (1979)] and Ray [Phys. Rev. A 26, 729 (1982)]. While the method is found to work well for a restricted class of harmonic potentials, its limitations in the case of anharmonic potentials are pointed out.