### Abstract

We study the semiclassical limit for bound states of the Hydrogen atom Hamiltonian H(ℏ) = - ℏ^{2}/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.

Original language | English (US) |
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Pages (from-to) | 316-340 |

Number of pages | 25 |

Journal | Helvetica Physica Acta |

Volume | 72 |

Issue number | 5-6 |

State | Published - Dec 1 1999 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Helvetica Physica Acta*,

*72*(5-6), 316-340.