Quantum Systems with Strong Constraining Potentials

Claudio Castelnovo

Kenneth S. Schmitz

Kevin Mitchell

We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is closely related to the geometric phase originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.

V. Pažma

Paolo Maraner

We discuss the motion of a quantum-mechanical system constrained to move on an arbitrary submanifold M of its configuration space R n by a real confining potential. A complete perturbative expansion for the Hamiltonian describing the dynamics of the system is obtained in terms of the quantities characterizing the intrinsic and extrinsic geometric properties of the constraint's surface M. In accordance with Heisenberg's principle the zeroth-order term of the expansion represents strong fluctuations of the system in directions normal to M, whereas the first-order term is naturally interpreted as the Hamiltonian describing the effective dynamics along the constraint's surface. The effective motion on M results in coupling with Abelian/non-Abelian gauge fields and quantum potentials. The rest of the perturbative expansion allows one to take into account further interactions between normal and effective degrees of freedom. As a concrete example we consider the dynamics of an electron constrained on a circle by a hypothetical semiconductor device. Dunham's expansion for the rotovibrational energy of a rigid diatom is also obtained.

Yoshihiko Abe, Ryo Nagai

We propose Quantum Constrained Hamiltonian Descent (QCHD), a continuous-time quantum framework for optimization under smooth equality constraints. The unconstrained counterpart, Quantum Hamiltonian Descent (QHD), minimizes a cost function by simulating the Schrödinger dynamics of a particle in the potential given by the cost function. Applying Dirac's formalism to the classical dynamics with constraints, we construct an intrinsic quantum Hamiltonian with a Laplace--Beltrami operator on the constraint manifold in a kinetic term. Its evolution preserves the constraints exactly without penalty terms and is expected to solve quantum-dynamically the constrained optimization problem. We analyze its relation to Quantum Hamiltonian Descent supplemented by a strong penalty term. For a squared-distance penalty, the effective tangential dynamics acquires an extrinsic-curvature potential and, in nontrivial normal-rotation sectors, a coupling to the normal-bundle connection. These embedding-dependent terms remain in the strong-penalty limit, generally yielding quantum dynamics different from QCHD. Numerical comparisons for circular, elliptical, and two simultaneous constraints show faster optimization and improved concentration near global minima for the initial states and finite penalty strengths considered.