Transition Matrix Elements for Large Momentum or Energy Transfer

Abstract

The radial matrix element $M=\int 0^{\infty }{R}_{k{l}^{\prime }}\left(\mathcal{r}\right)j_L\left(q\mathcal{r}\right)R_{\mathrm{nl}}\left(\mathcal{r}\right)r^{2}d\mathcal{r}$, which appears in collision theory and photon absorption ($L=1\mathrm{}$, $q\sim 0$), 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 $M\propto q^{-(l+l^{\prime }+4)}$ and for large $k$ and normalization per unit energy, $M\propto k^{-(l+L+\frac{7}{2})}{q}^L$. The asymptotic trend of Altshuler's equivalent forms of the matrix elements is discussed.