99-69 George A. Hagedorn Sam L. Robinson
Approximate Rydberg States of the Hydrogen Atom that are Concentrated near Kepler Orbits (1250K, latex with 4 ps figures) Mar 3, 99
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Abstract. We study the semiclassical limit for bound states of the Hydrogen atom Hamiltonian $$H(\hbar)\,=\,-\,\frac {\hbar^2}2\,\Delta\,-\,\frac 1{|x|}.$$ For each Kepler orbit of the corresponding classical system, we construct a lowest order quasimode $\Psi(\hbar,x)$ for $H(\hbar)$ when the appropriate Bohr-Sommerfeld conditions are satisfied. This means that $\Psi(\hbar,x)$ is an approximate solution of the Schr\"{o}dinger equation in the sense that $$\left\|\,\left[ H(\hbar)-E(\hbar)\right]\,\Psi(\hbar,\cdot)\,\right\|\,\leq\, C\,\hbar^{3/2}\,\left\|\Psi(\hbar,\cdot)\right\| .$$ The probability density $|\Psi(\hbar,\,x)|^2$ is concentrated near the Kepler ellipse in position space, and its Fourier transform has probability density $|\widehat{\Psi}(\hbar,\,\xi)|^2$ concentrated near the Kepler circle in momentum space. Although the existence of such states has been demonstrated previously, the ideas that underlie our time-dependent construction are intuitive and elementary.

Files: 99-69.src( 99-69.keywords , sam4.tex , pcircular.ps , pelliptical.ps , xcircular.ps , xelliptical.ps )