The generation of rotational bands by deep, diffuse potentials

Numerical integration of the single-particle Schrödinger equation containing a deep, diffuse, spherically symmetric, attractive, local potential often gives rise to bound-state energy eigenvalues, which lie on almost perfect rotational bands. Each band is characterized by a constant value of (2n + l), where n is the number of interior nodes in the radial wave function and l is the orbital angular-momentum quantum number of a member state. The constants of near-proportionality between the state energies and l(l + 1) for the various bands are considered. An expression for them derived from the Bohr–Sommerfeld quantization rule is found to give numerical results in excellent agreement with those obtained by direct numerical integration of the Schrödinger equation. However, no analytic expressions could be obtained from it, except in the most trivial cases. Also, an approximate expression for these rotational parameters is derived using the Rayleigh–Schrödinger perturbation theory up to fourth order for the case of any potential that may be expanded in ascending even powers of the radial coordinate, r, for which the perturbation expansion converges. This approximation is excellent if (2n + l + 3/2)[Formula: see text], where m is the mass appearing in the Schrödinger equation, V0 is the depth of the attractive potential, and a is a length parameter characterizing its diffuseness. The general results are illustrated with applications to some deep attractive Gaussian potentials of various widths.