Quantization of the double pendulum

Ali, M. K.

doi:10.1139/p96-040

Cited by 10 publications

(14 citation statements)

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“…Namely, different results were obtained depending whether the constraints were imposed before or after the quantization. Another example is yet unsolved problem of quantization of the double pendulum [2]. We also recall the troublesome questions [3] arising in the analysis of the Heisenberg uncertainty relations arising in the case when one of observables, as for instance the angle operator, has the compact spectrum.…”

Section: Introductionmentioning

confidence: 99%

Coherent states for a quantum particle on a circle

Kowalski

Rembieliński

Papaloucas

1996

J. Phys. A: Math. Gen.

124

224

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Section: Introductionmentioning

confidence: 99%

Coherent states for a quantum particle on a circle

Kowalski

Rembieliński

Papaloucas

1996

J. Phys. A: Math. Gen.

124

224

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“…The ambiguity here has a clear physical meaning: to determine the true quantum mechanics one needs to know the mechanism through which the particle is bound to the surface of constraint; it is not sufficient to know only the intrinsic properties of the surface itself. Similarly, there is no "correct" answer to the problem of quantizing a classical double pendulum [4]: quantum dynamics at O(h 2 ) is determined by the precise way in which one takes to infinity the rigidity of the two rods.In a two-dimensional system, the energy spacing between adjacent levels is O(h 2 ), i.e. of the same order as the quantization ambiguity demonstrated above.…”

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confidence: 99%

Quantization ambiguity, ergodicity and semiclassics

Kaplan

2002

New J. Phys.

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A simple argument shows that eigenstates of a classically ergodic system are individually ergodic on coarse-grained scales. This has implications for the quantization ambiguity in ergodic systems: the difference between alternative quantizations is suppressed compared with the O(h 2 ) ambiguity in the integrable case. For two-dimensional ergodic systems in the high-energy regime, individual eigenstates are independent of the choice of quantization procedure, in contrast with the regular case, where even the ordering of eigenlevels is ambiguous. Surprisingly, semiclassical methods are shown to be much more precise for chaotic than for integrable systems.For many years, it has been widely recognized that "quantizing" a given classical system is inherently an ambiguous procedure, as a large family of quantum Hamiltonians may have the same classical limit [1]. For example, given the classical dynamics of a particle constrained to move on a closed loop, different choices of boundary condition give rise to different phases relating classical paths of different winding number. Knowledge of these Aharonov-Bohm phases is of course necessary to construct a semiclassical dynamics (which includes interference between classical paths), and thus many semiclassical theories correspond to the same classical dynamics. Physically, this O(h) or gauge ambiguity may be associated with the possibility of varying the magnetic flux enclosed by the loop. This is not all, however: there are also many quantum theories, differing at O(h 2 ) or higher in the Hamiltonian, which all have the same semi-classical limit. There are many ways of seeing this O(h 2 ) ambiguity; one of the simplest is to imagine making a canonical transformation on the classical phase space, applying the canonical quantization prescription to the new coordinates, and then transforming back to the original coordinate system. Generically, one then obtains a new quantum Hamiltonian which differs from the original by O(h 2 ) plus higher order terms:Ĥwhere the operatorÂ has a well-defined classical limit [2]. As classical dynamics is of course independent of the choice of coordinate system, this implies that quantization is inherently ambiguous at second order inh. An even more striking case is that of a particle constrained to move on a two-dimensional surface embedded * kaplan@physics.harvard.edu in three-dimensional space. Here, the non-trivial metric contained in the kinetic term gives rise to obvious operator-ordering ambiguities in canonical quantization; this has led to much discussion in the literature over whether a term proportional to the local Gaussian curvature R of the surface should be added to the Hamiltonian, and if so, what the proportionality constant should be (different prescriptions suggestingh 2 R/8,h 2 R/6, and h 2 R/12 as the "correct" answer) [3][4][5]. In the path integral approach, ambiguities at the same order arise in choosing how to incorporate the metric into the kernel and in deciding at what point in the infinitesimal time interval to eval...

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“…The author of Ref. [5] points out that the ambiguity related to the choice of λ could only be settled using the experimental data obtained for the particular system in question. This point of view was adopted independently in Ref.…”

Section: Introductionmentioning

confidence: 99%

Scattering due to geometry: Case of a spinless particle moving on an asymptotically flat embedded surface

2018

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A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.

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Quantization of the double pendulum

Cited by 10 publications

References 0 publications

Coherent states for a quantum particle on a circle

Coherent states for a quantum particle on a circle

Quantization ambiguity, ergodicity and semiclassics

Scattering due to geometry: Case of a spinless particle moving on an asymptotically flat embedded surface

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