Hydrogen relative-motion equation

Abstract. Cross sections for electron transfer in collisions of atomic hydrogen with fully stripped carbon ions are studied for impact energies from 0.1 to 500 keV/u. A semi-classical close-coupling approach is used within the impact parameter approximation. To solve the time-dependent Schrodinger equation the electronic wave function is expanded on a two-center atomic state basis set. The projectile states are modified by translational factors to take into account the relative motion of the two centers. For the processes C6++H(1.s) —> C5+ (nlm) + H+, we present shell-selective electron transfer cross sections, based on computations performed with an expansion spanning all states ofC5+( =l-6) shells and the H(ls) state. 

This equation can be immediately separated into two, one of which represents the translational motion of the molecule as a whole and the other the relative motion of the two particles. In fact, this separation can be accomplished in a somewhat more general case, namely, when the potential energy V is a general function of the relative positions of the two particles, that is, V = F(ss — xi, 2/2 — yi, zi — i). This includes, for example, the hydrogen atom in a constant electric field, the potential... 

The time-independent Schrodinger equation for a hydrogen atom was separated into a one-particle Schrodinger equation for the motion of the center of mass of the two particles and a one-particle Schrodinger equation for the motion of the electron relative to the nucleus. The motion of the center of mass is the same as that of a free particle. The Schrodinger equation for the relative motion was solved by separation of variables in spherical polar coordinates, assuming the trial function... 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

The Schrodinger equation for the hydrogen atom can he separated into one equation for the motion of the center of mass and one equation for the motion of the electron relative to the nucleus. 

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