Spacings of Quarkonium Levels with the Same Principal Quantum Number

Abstract

The spacings between bound-state levels of the Schrödinger equation with the same principal quantum number $N$ but orbital angular momenta $\ell$ differing by unity are found to be nearly equal for a wide range of power potentials $V = λr^ν$, with $E_{N \ell} \approx F(ν, N) - G(ν,N) \ell$. Semiclassical approximations are in accord with this behavior. The result is applied to estimates of masses for quarkonium levels which have not yet been observed, including the 2P $c \bar c$ states and the 1D $b \bar b$ states.