The Inverse-Cube Central Force Field in Quantum Mechanics

Abstract

The problem of the motion of a particle in an inverse-cube central force field is fully treated by quantum mechanics and the results compared with the classical theory. Taking the effective radial potential energy as $\frac{S}{r^2}$, although the solutions for negative energy for $0\geqq S\geqq \frac{-h^2}{32{\pi }^2\mu }$ satisfy the usual boundary conditions, they can not be admitted because the Hamiltonian is not Hermitian in these solutions. This corresponds to taking ${(l+\frac{1}{2})}^2$ in place of $l(l+1\mathrm{})\mathrm{}$ as the analogue of the square of the classical angular momentum. If we do this, we get a complete analogy between the classical and quantum mechanically allowed solutions, with no quantization. The solutions involve Bessel functions of both real and imaginary orders with both real and imaginary arguments.