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.
|Number of pages||11|
|Journal||Journal of Mathematical Physics|
|Publication status||Published - 1 Dec 1982|
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics