### Abstract

We study one-dimensional quantum mechanical systems in the semiclassical limit. We construct a lowest order quasimode ψ(ℏ) for the Hamiltonian H(ℏ) when the energy E and Planck's constant ℏ satisfy the appropriate Bohr - Sommerfeld conditions. This means that ψ(ℏ) is an approximate solution of the Schrödinger equation in the sense that \\[H(ℏ) - E]ψ(ℏ)|| ≤ Cℏ^{3/2}||ψ(ℏ)||. It follows that H(ℏ) has some spectrum within a distance Cℏ^{3/2} of E. Although the result has a long history, our time-dependent construction technique is novel and elementary.

Original language | English (US) |
---|---|

Pages (from-to) | 10113-10129 |

Number of pages | 17 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 31 |

Issue number | 50 |

DOIs | |

State | Published - Dec 18 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

*Journal of Physics A: Mathematical and General*,

*31*(50), 10113-10129. https://doi.org/10.1088/0305-4470/31/50/009

**Bohr - Sommerfeld quantization rules in the semiclassical limit.** / Hagedorn, George A.; Robinson, Sam L.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 31, no. 50, pp. 10113-10129. https://doi.org/10.1088/0305-4470/31/50/009

}

TY - JOUR

T1 - Bohr - Sommerfeld quantization rules in the semiclassical limit

AU - Hagedorn, George A.

AU - Robinson, Sam L

PY - 1998/12/18

Y1 - 1998/12/18

N2 - We study one-dimensional quantum mechanical systems in the semiclassical limit. We construct a lowest order quasimode ψ(ℏ) for the Hamiltonian H(ℏ) when the energy E and Planck's constant ℏ satisfy the appropriate Bohr - Sommerfeld conditions. This means that ψ(ℏ) is an approximate solution of the Schrödinger equation in the sense that \\[H(ℏ) - E]ψ(ℏ)|| ≤ Cℏ3/2||ψ(ℏ)||. It follows that H(ℏ) has some spectrum within a distance Cℏ3/2 of E. Although the result has a long history, our time-dependent construction technique is novel and elementary.

AB - We study one-dimensional quantum mechanical systems in the semiclassical limit. We construct a lowest order quasimode ψ(ℏ) for the Hamiltonian H(ℏ) when the energy E and Planck's constant ℏ satisfy the appropriate Bohr - Sommerfeld conditions. This means that ψ(ℏ) is an approximate solution of the Schrödinger equation in the sense that \\[H(ℏ) - E]ψ(ℏ)|| ≤ Cℏ3/2||ψ(ℏ)||. It follows that H(ℏ) has some spectrum within a distance Cℏ3/2 of E. Although the result has a long history, our time-dependent construction technique is novel and elementary.

UR - http://www.scopus.com/inward/record.url?scp=0040113664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040113664&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/31/50/009

DO - 10.1088/0305-4470/31/50/009

M3 - Article

AN - SCOPUS:0040113664

VL - 31

SP - 10113

EP - 10129

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 50

ER -