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)
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.