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arXiv:hep-th/9712052 · Journal of Mathematical Physics

Constrained dynamics for quantum mechanics. I. Restricting a particle to a surface

Hendrik Grundling, C. A. Hurst

Published Jun 1, 1998• 14 citations
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Abstract

We analyze constrained quantum systems where the dynamics do not preserve the constraints. This is done, in particular, for the restriction of a quantum particle in Rn to a curved submanifold, and we propose a method of constraining and dynamics adjustment that produces the right Hamiltonian on the submanifold when tested on known examples. This method will be the germ of a “Dirac algorithm for quantum constraints.” We generalize it to the situation where the constraint is a general self-adjoint operator with some additional structures.

DOI
10.1063/1.532241

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We analyze constrained quantum systems where the dynamics do not preserve the constraints. This is done, in particular, for the restriction of a quantum particle in Rn to a curved submanifold, and we propose a method of constraining and dynamics adjustment that produces the right Hamiltonian on the submanifold when tested on known examples. This method will be the germ of a “Dirac algorithm for quantum constraints.” We generalize it to the situation where the constraint is a general self-adjoint operator with some additional structures.
Using the projection operator method, the quantization of a dynamical system constrained to a submanifold MN−1 embedded in the Euclidean space RN is investigated. The resultant Hamiltonian contains the quantum correction terms which are missed in the usual approach with Dirac's formalism. Such correction terms are investigated from a geometrical point of view in the case that the system is constrained to a 2-dimensional orientatable compact curved spaces M2 embedded in R3. It is shown that the curvatures of these submanifolds are closely related to the quantum correction terms.
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.
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 framework for deriving equations of motion for constrained quantum systems is introduced and a procedure for its implementation is outlined. In special cases, the proposed new method, which takes advantage of the fact that the space of pure states in quantum mechanics has both a symplectic structure and a metric structure, reduces to a quantum analogue of the Dirac theory of constraints in classical mechanics. Explicit examples involving spin- particles are worked out in detail: in the first example, our approach coincides with a quantum version of the Dirac formalism, while the second example illustrates how a situation that cannot be treated by Dirac's approach can nevertheless be dealt with in the present scheme.
A framework for deriving equations of motion for constrained quantum systems is introduced and a procedure for its implementation is outlined. In special cases, the proposed new method, which takes advantage of the fact that the space of pure states in quantum mechanics has both a symplectic structure and a metric structure, reduces to a quantum analogue of the Dirac theory of constraints in classical mechanics. Explicit examples involving spin- particles are worked out in detail: in the first example, our approach coincides with a quantum version of the Dirac formalism, while the second example illustrates how a situation that cannot be treated by Dirac's approach can nevertheless be dealt with in the present scheme.
In this paper, we present a detailed review/analysis of the Dirac quantisation of Hamiltonian systems with constraints. To this end, we use, as a guide, the physical example provided by the dynamics of a solid ball rolling, without slipping, down an inclined plane under the action of gravity. After all, however simple this physical system may be, it provides a rich framework for this analysis since, in addition to allowing us to discuss scenarios involving holonomic and non-holonomic constraints, it is also a gauge system. Indeed, due to this latter fact, we have carefully detailed how the transition, from classical to quantum mechanics, must be guided by the Dirac-Bergmann algorithm and by the consequent replacement of Dirac brackets with commutators. As a central result, we demonstrate that the restriction of the Hamiltonian operator of this system with constraints to the physical Hilbert subspace (which is identified with the quantisation of these constraints) reproduces the same Schrödinger equation that can be originally obtained in intrinsic terms, a fact that only reinforces the consistency of the Dirac quantisation method.