A COMPLETE PERTURBATIVE EXPANSION FOR QUANTUM-MECHANICS WITH CONSTRAINTS

We discuss the motion of a quantum-mechanical system constrained to mo ve on an arbitrary submanifold M of its configuration space R(n) by a real confining potential. A complete perturbative expansion for the Ha miltonian describing the dynamics of the system is obtained in terms o f the quantities characterizing the intrinsic and extrinsic geometric properties of the constraint's surface M. In accordance with Heisenber g's principle the zeroth-order term of the expansion represents strong fluctuations of the system in directions normal to M, whereas the fir st-order term is naturally interpreted as the Hamiltonian describing t he effective dynamics along the constraint's surface. The effective mo tion on M results in coupling with Abelian/nonAbelian gauge fields and quantum potentials. The rest of the perturbative expansion allows one to take into account further interactions between normal and effectiv e degrees of freedom. As a concrete example we consider the dynamics o f an electron constrained on a circle by a hypothetical semiconductor device. Dunham's expansion for the rotovibrational energy of a rigid d iatom is also obtained.