Equivalence of the Sudden Approximation to the High-Energy Limit of the First Born Approximation

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Abstract
The ratio of total cross sections for two different inelastic events in a given target as predicted by the sudden approximation is shown to be equal to the high-energy limit of the corresponding first Born approximation to the ratio. This relationship between the two theories is established for the special cases of the ratio of single to double ionization of helium atoms by electrons and by hydrogen atoms. The electron-helium-atom ratio is found to be the same as the hydrogen-atom-helium-atom ratio. The proof supposes that the same approximate wave functions were used in both theories and that the dominant term in the high-energy expansion of the first Born total cross section is independent of energy-conservation requirements. This latter point is established for a general ionization event.