A rigorous theory of the s-wave elastic scattering of electrons from hydrogen is presented. The Schrodinger equation is reduced to an infinite set of coupled, two-dimensional partial differential equations. A zeroth order scattering problem is defined by neglecting the coupling terms of the first equation. An exact relation is derived between the phase shift of this zeroth order problem and the true phase shift. The difference between these is contained in a rapidly convergent series whose terms correspond adiabatically to multipole distortions of the hydrogen by the incoming electron. The physical significance of the zeroth order problem is discussed, and its recognition is considered basic to the understanding of the scattering problem. The exchange approximation for s-wave scattering is shown to be a variational approximation of the zeroth order problem. A perturbation theory is introduced to calculate the higher order corrections. The dipole correction has an increasingly important quantitative effect in the limit of zero energy. The effect of the long-range part of this correction on the scattering length can be expressed by a formula in terms of inverse powers of a long-range parameter E. Phase shifts for both singlet and triplet scattering are calculated, including up to quadrupole terms. The convergence is such that this number of terms should yield better than four-place accuracy. Uncertainties in our calculated values decrease the accuracy to approximately three significant figures. I. INTRODUCTIO¹HE elastic scattering of electrons from atomic hydrogen is the most fundamental three-body scattering problem of quantum mechanics. Nevertheless, the process of theoretical understanding has been slow and is still not complete. Specifically, the lack of a real quantitative understanding of this problem, as compared with its counterpart, the ground state of helium problem, can be traced to two causes. The first was the lack of a minimum principle, which guarantees that not only is a certain quantity variational in character, but more important that it is greater than (or less than) the exact quantity one wants to evaluate. The lack of a minimum principle has recently been overcome by Rosenberg, Spruch, and O' Malley, who have derived minimum principles for the scattering length' and who are attempting to generalize the procedure to include phase shifts as well. ' The second aspect of scattering problems which has slowed the process of quantitative understanding is simply the fact that the scattering parameters are more sensitive functions of the wave function than, say, the ground-state energy. The sensitivity may be illustrated in the case of the singlet scattering of electrons by hydrogen, where there is known to be a singlet bound state of the H ion with a binding energy of 14.460 ev. The electron amenity, e, is defined as the difference between this energy and that of a hydrogen atom and a free electron. Clearly, this is the quantity of physical significance, for only when it is negative is the H bound. The...
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