Calculation of cross sections for electron capture between arbitrary hydrogenic states of target and projectile

We present a method for the calculation of cross sections for electron capture from an arbitrary $n$,$l$,$m$ shell of a hydrogenic target atom into an arbitrary $n^{\prime }$,$l^{\prime }$,$m^{\prime }$ shell of a fast hydrogenic projectile in the continuum-intermediate-state approximation. We reduce the scattering amplitude to a one-dimensional integral in real space. The numerical results are presented for the reaction ${\mathrm{H}}^{+}+\mathrm{H}\mathrm{}\left(2l\right)\mathrm{}\to \mathrm{H}\left({3l}^{\prime }\right)+{\mathrm{H}}^{+}$. Comparison is made with a recent eikonal calculation.