 9969 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 BohrSommerfeld 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 timedependent construction are
intuitive and elementary.
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