Approximate Rydberg states of the hydrogen atom that are concentrated near Kepler orbits

We study the semiclassical limit for bound states of the Hydrogen atom Hamiltonian H(ℏ) = - ℏ²/2 Δ - 1/|x|. For each Kepler orbit of the corresponding classical system, we construct a lowest order quasimode ψ(ℏ,x) for H(ℏ) when the appropriate Bohr-Sommerfeld conditions are satisfied. This means that ψ(ℏ, x) is an approximate solution of the Schrödinger equation in the sense that ∥ [H(ℏ) - E(ℏ)] ψ(ℏ, ·) ∥ ≤ C ℏ^3/2 ∥ψ(ℏ, ·)∥. The probability density |ψ(ℏ, x)|² is concentrated near the Kepler ellipse in position space, and its Fourier transform has probability density |ψ̂(ℏ, ξ)|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.