Physical Review Letters

From the early days of quantum mechanics, it has long been expected that the greater the quantum number for a certain degree of freedom (corresponding to a greater number of nodes in the corresponding wave function), the better the semiclassical approximation [1]. This expectation, which is often referred to as Bohr’s correspondence principle, has persisted despite a lack of rigorous proof for an arbitrary potential. It works well in Coulombic systems and has been assumed to be a general principle partly because for systems in which the asymptotic interaction is of the form of 1 rn with n . 2, the only systematic understanding of the highly excited bound spectra has been based on a semiclassical consideration [2] and there has been no corresponding quantum theory to compare with [3]. Stimulated by advances in cold-atom collisions, especially the development of photoassociative spectroscopy (see, e.g., [4–14]), considerable progress has been made in the understanding of the excited spectra of systems in which the asymptotic interaction is not Coulombic [4– 21]. In particular, analytic solutions of the Schrödinger equation for 1 r6 and 1 r3 potentials have been obtained [18–21]. By comparing the fully quantum result based upon these solutions with the corresponding semiclassical predication and also by examining the criterion for the applicability of the semiclassical approximation, we show the expectation that semiclassical results work better for more highly excited states is not applicable to systems in which the asymptotic potential is of the form of 2Cn rn with n . 2. In fact, we will show for such systems that the opposite is true, i.e., the semiclassical approximation