Canonical quantization of motion on submanifolds

Golovnev, Alexey

doi:10.1016/s0034-4877(09)90020-9

Cited by 17 publications

(21 citation statements)

References 37 publications

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“…The following is a list of various forms of the GM in the current literature. Because (6)- (8) give the most general GM of p (α,β)i satisfying the fundamental commutators (20 ) (v) The fifth choice is made by many groups based on different theoretical grounds and it should be α = 1 and β = 0 13,17,[19][20][21][22][23]30 .…”

Section: Geometric Momenta For a Particle On The Sphere: A Reviewmentioning

confidence: 99%

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Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

2011

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In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum p θ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as p i (i = 1, 2, 3) are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between p i generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.

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Section: Geometric Momenta For a Particle On The Sphere: A Reviewmentioning

confidence: 99%

“…The terms on the right-hand sides of these equations must be all zero; otherwise, relation (23) as [x i , H] = i p (α,β)i /µ will be violated. The only solution is simply…”

Section: Complete Determination Of Geometric Momenta and The Hammentioning

confidence: 99%

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

2011

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“…(2) shows that in general the curvature coefficients, λ 1 and λ 2 , depend on the details of the confining forces [24], i.e., their values cannot be determined from first principle. Viewing the system as a second-class constrained system and employing Dirac's quantization program for such systems leads to the same conclusion; the Hamiltonian has the form (2), but the curvature coefficients cannot be uniquely determined from theoretical considerations [25]. These observations provide further motivation for the empirical determination of the curvature coefficients by performing scattering experiments.…”

Section: Introductionmentioning

confidence: 89%

Geometric scattering in the presence of line defects

Bui,

Mostafazadeh,

Seymen

2020

Preprint

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A non-relativistic scalar particle moving on a curved surface undergoes a geometric scattering whose behavior is sensitive to the theoretically ambiguous values of the intrinsic and extrinsic curvature coefficients entering the expression for the quantum Hamiltonian operator. This suggests using the scattering data to settle the ambiguity in the definition of the Hamiltonian. It has recently been shown that the inclusion of point defects on the surface enhances the geometric scattering effects. We perform a detailed study of the geometric scattering phenomenon in the presence of line defects for the case that the particle is confined to move on a Gaussian bump and the defect(s) are modeled by delta-function potentials supported on a line or a set of parallel lines normal to the scattering axis. In contrast to a surface having point defects, the scattering phenomenon associated with this system is generically geometric in nature in the sense that for a flat surface the scattering amplitude vanishes for all scattering angles θ except θ = θ 0 and π − θ 0 , where θ 0 is the angle of incidence. We show that the presence of the line defects amplifies the geometric scattering due to the Gaussian bump. This amplification effect is particularly strong when the center of the bump is placed between two line defects.

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“…Note that in these calculations (23), (24) and (25) where the constraints are of second-class, we need to deal with H instead of H p for we have,ḟ…”

Section: Dirac's Theory For Systems Of Second-class Constraintsmentioning

confidence: 99%

Geometric Momentum in the Monge Parametrization of Two-Dimensional Sphere

Xun

Liu

2013

Int. J. Geom. Methods Mod. Phys.

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A two dimensional surface can be considered as three dimensional shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of two dimensional sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = i δij − xixj/r 2 rather than [xi, pj] = i δij that does not hold true any more. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

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Canonical quantization of motion on submanifolds

Cited by 17 publications

References 37 publications

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

Geometric scattering in the presence of line defects

Geometric Momentum in the Monge Parametrization of Two-Dimensional Sphere

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