Approximate rydberg states of the hydrogen atom that are concentrated nearKepler orbits

We study the semiclassical limit for bound states of the Hydrogen atom Hami ltonian H(h) = -h(2)/2 Delta - 1/\x\ For each Kepler orbit of the correspon ding classical system, we construct a lowest order quasimode psi(h, x) for H(h) when the appropriate Bohr-Sommerfeld conditions are satisfied. This me ans that psi(h,x) is an approximate solution of the Schrodinger equation in the sense that \\[H(h) - E(h)] psi(h(1))\\ less than or equal to Ch(3/2) \ \psi(h,)\\. The probability density \psi(h, x)\(2) is concentrated near the Kepler ellipse in position space, and its Fourier transform has probabilit y density \psi(h, xi)\(2) concentrated near the Kepler circle in momentum s pace. Although the existence of such states has been demonstrated previousl y, the ideas that underlie our time-dependent construction are intuitive an d elementary.