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

摘要

We study the semiclassical limit for bound states of the Hydrogen atom Hamiltonian H(h) = -h(2)/2 Delta - 1/x For each Kepler orbit of the corresponding classical system, we construct a lowest order quasimode psi(h, x) for H(h) when the appropriate Bohr-Sommerfeld conditions are satisfied. This means 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 probability density psi(h, 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. [References: 14]