### 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) |
---|---|

Pages (from-to) | 316-340 |

Number of pages | 25 |

Journal | Helvetica Physica Acta |

Volume | 72 |

Issue number | 5-6 |

State | Published - Dec 1 1999 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Helvetica Physica Acta*,

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

**Approximate Rydberg states of the hydrogen atom that are concentrated near Kepler orbits.** / Hagedorn, George A.; Robinson, Sam L.

Research output: Contribution to journal › Article

*Helvetica Physica Acta*, vol. 72, no. 5-6, pp. 316-340.

}

TY - JOUR

T1 - Approximate Rydberg states of the hydrogen atom that are concentrated near Kepler orbits

AU - Hagedorn, George A.

AU - Robinson, Sam L

PY - 1999/12/1

Y1 - 1999/12/1

N2 - 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 ℏ3/2 ∥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.

AB - 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 ℏ3/2 ∥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.

UR - http://www.scopus.com/inward/record.url?scp=0008524636&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0008524636&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0008524636

VL - 72

SP - 316

EP - 340

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 5-6

ER -