Transition Matrix Elements for Large Momentum or Energy Transfer

The radial matrix element M=∫0^∞R_kl^'(r)jL(qr)Rₙₗ(r)r²dr, which appears in collision theory and photon absorption ($L=1$, q~0), has been studied in the past for hydrogenic wave functions. Its behavior for large q or large k is now shown to depend only on the expansion of the wave functions near the nucleus and on an application of selection rules. For large q, the trend is M∝q^-(l+l^'+4) and for large k and normalization per unit energy, M∝k^-(l+L+7/2)qL. The asymptotic trend of Altshuler's equivalent forms of the matrix elements is discussed.