Energy levels and level ordering in the WKB approximation

Abstract

Energy levels and level orderings for a particle in a non-relativistic potential are examined in the WKB approximation. In particular, power-law potentials ( V( r) = ar γ , −2 < γ < ∞) are discussed in some detail. The energy levels are shown to be determined in terms of a single function G( η, γ) of a variable η. Expansions of this function, valid for small (large) angular momentum quantum numbers ( l) and large (small) radial quantum numbers ( n), approximate the energy levels well. The ordering of the levels follows from the monotonic behavior of ( ∂/ ∂η) G( η, γ). The values γ = 2 (harmonic oscillator potential) and γ = −1 (Coulomb potential) for which the WKB approximation gives the exact (i.e. Schrödinger) results lead to degenerate levels. It is about these values of γ that the monotonic behavior of ( ∂/ ∂η) G( η, γ) changes sign (as a function of γ). We also find an ordering theorem for arbitrary central potentials which is valid for large l and small n and is possibly correct for smaller l. The ordering depends on various sums of derivatives of the potential. Similar theorems, which follow from the Schrödinger equation, have been obtained recently for low-lying levels and are compared to our results.