### Abstract

The systems-theoretic concept of controllability is elaborated for quantum-mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space ℋ of the system. The Schrödinger equation for a quantum object influenced by adjustable external fields provides a state-evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain script D_{ω} in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite-dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of script D_{ω} with a suitably defined finite-dimensional submanifold of the unit sphere S_{script K} in ℋ. Several soluble examples are presented to illuminate the general theoretical results.

Original language | English |
---|---|

Pages (from-to) | 2608-2618 |

Number of pages | 11 |

Journal | Journal of Mathematical Physics |

Volume | 24 |

Issue number | 11 |

Publication status | Published - 1 Dec 1982 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*24*(11), 2608-2618.

**On the controllability of quantum-mechanical systems.** / Huang, Garng Morton; Tarn, T. J.; Clark, John W.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 24, no. 11, pp. 2608-2618.

}

TY - JOUR

T1 - On the controllability of quantum-mechanical systems

AU - Huang, Garng Morton

AU - Tarn, T. J.

AU - Clark, John W.

PY - 1982/12/1

Y1 - 1982/12/1

N2 - The systems-theoretic concept of controllability is elaborated for quantum-mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space ℋ of the system. The Schrödinger equation for a quantum object influenced by adjustable external fields provides a state-evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain script Dω in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite-dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of script Dω with a suitably defined finite-dimensional submanifold of the unit sphere Sscript K in ℋ. Several soluble examples are presented to illuminate the general theoretical results.

AB - The systems-theoretic concept of controllability is elaborated for quantum-mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space ℋ of the system. The Schrödinger equation for a quantum object influenced by adjustable external fields provides a state-evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain script Dω in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite-dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of script Dω with a suitably defined finite-dimensional submanifold of the unit sphere Sscript K in ℋ. Several soluble examples are presented to illuminate the general theoretical results.

UR - http://www.scopus.com/inward/record.url?scp=36749114459&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36749114459&partnerID=8YFLogxK

M3 - Article

VL - 24

SP - 2608

EP - 2618

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 11

ER -