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arXiv:hep-th/9409080 · Journal of Physics A Mathematical and General

A complete perturbative expansion for quantum mechanics with constraints

Paolo Maraner

Published May 21, 1995• 59 citations
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Abstract

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.

DOI
10.1088/0305-4470/28/10/021

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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.
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.
When quantum mechanical motion of a particle constrained to a curved surface in the Euclidean space is considered, three different approaches may be adopted to describe the system. One is the confining approach which enforces a particle to stay in the surface by a physical confining potential. Both of the others follow Dirac's general prescription to treat a constrained system, but employ different constraints, namely the usual and the conservative constraints, respectively. The three approaches lead to results which do not in general agree with each other, though they give the same classical equation of motion. It is emphasized that the arbitrariness in the ordering of operators cannot be avoided when the latter two approaches are adopted. In the approach with the conservative constraint, however, there is a physical ordering, which leads to the result in agreement with that of the confining approach. We interpret these discrepancy and agreement in simple physical terms.
We derive the effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) in the asymptotic limit, where the restoring forces tend to infinity. In contrast to earlier works, we consider, at the same time, the effects of variations in the constraining potential and the effects of interior and exterior geometry, which appear at different energy scales and, thus, provide a complete picture, which ranges over all interesting energy scales. We show that the leading order contribution to the effective Hamiltonian is the adiabatic potential given by an eigenvalue of the confining potential well known in the context of adiabatic quantum waveguides. At next to leading order, we see effects from the variation of the normal eigenfunctions in the form of a Berry connection. We apply our results to quantum waveguides and provide an example for the occurrence of a topological phase due to the geometry of a quantum wave circuit (i.e., a closed quantum waveguide).
When quantum mechanical motion of a particle constrained to a curved surface in the Euclidean space is considered, three different approaches may be adopted to describe the system. One is the confining approach which enforces a particle to stay in the surface by a physical confining potential. Both of the others follow Dirac's general prescription to treat a constrained system, but employ different constraints, namely the usual and the conservative constraints, respectively. The three approaches lead to results which do not in general agree with each other, though they give the same classical equation of motion. It is emphasized that the arbitrariness in the ordering of operators cannot be avoided when the latter two approaches are adopted. In the approach with the conservative constraint, however, there is a physical ordering, which leads to the result in agreement with that of the confining approach. We interpret these discrepancy and agreement in simple physical terms.
A realistic analysis shows that constraining a quantomechanical system produces the effective dynamics to be coupled with abelian/nonabelian gauge fields and quantum potentials induced by the intrinsic and extrinsic geometrical properties of the constraint’s surface. This phenomenon is observable in the effective rotational motion of some simple polyatomic molecules. By considering specific examples it is shown that the effective Hamiltonians for the nuclear rotation of linear and symmetric top molecules are equivalent to that of a charged system moving in a background magnetic-monopole field. For spherical top molecules an explicit analytical expression of a non-abelian monopole-like field is found. Quantum potentials are also relevant for One of the first questions addressed by researchers who first studied the properties of surfaces embedded in the three-dimensional euclidean space, at