The virial theorem for electron capture

The virial theorem for the capture of an electron from a hydrogenic atomic system by a fully stripped atomic ion is derived using a Schwinger variational method together with an impact-parameter time-dependent relativistic approach. This requires the operator in the time-dependent differential equation describing the collision to be Hermitian and it is shown that this condition corresponds to discarding all terms arising from the absorption into and out of states not included in the expansion of the Schrodinger wavefunction. It is shown that the virial theorem for electron capture can also be derived without the use of the variational method and this is done for the scattering amplitude with particular reference to the first-order approximation. The relationship between the impact-parameter approach and the time-independent wave treatment of the virial theorem for the scattering amplitude is examined.