Geometric Momentum and a Probe of Embedding Effects

Liu, Q. H.

doi:10.7566/jpsj.82.104002

Cited by 22 publications

(13 citation statements)

References 42 publications

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“…However, as was noted in Ref. [26] it has zero projection along this normal direction. In the present work we show how to trade off the M û3 component of this momentum for an expression (Eq.…”

Section: Introductionsupporting

confidence: 61%

“…This is in contrast to the form p s = −i (∇ ′ + M û3 ), which has an "apparent" radial projection, which is not the case, however in Ref. [26]. Moreover, the fact that one can trade off the presence of the mean curvature in this kinematical surface momentum for a specific symmetric expression of the derivatives suggests that the appearance of the curvature in the momentum is actually a way of dictating this specific symmetric expression upon confining to the surface.…”

Section: The Symmetric Form Of the Hermitian Surface Momentummentioning

confidence: 89%

“…The same applies for the case of a cylindrical surface. Closing, we note that in the work, [26] an expression (Eq. ( 20) in the reference) for the quantum centripetal force valid for surfaces with constant curvatures (like the cylinder and sphere) was suggested by making analogy with the classical expression and symmetrizing.…”

Section: The Symmetric Form Of the Hermitian Surface Momentummentioning

confidence: 99%

“…The same expression was derived by us within the framework of the thin-layer quantization. [26] This momentum, despite being the momentum operator on the surface has an "apparent" component normal to the surface. However, as was noted in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…[27] that for surfaces with constant curvature, the expression for the quantum centripetal force is the same as the classical upon applying a simple symmetrizing in order to take care of operator ordering issues. In a more recent work, [26] we have obtained an expression for the centripetal force of a particle on a cylindrical and spherical surface that although "apparently" not radial, was checked to have zero projection along the surface and a radial component formally similar to the classical expression. Here, we derive, starting from the symmetrized momentum and applying the Heisenberg equations of motion an expression for the centripetal force for a particle on the surface of a cylinder and a sphere that is just a symmetrization of the classical expression of this force, namely, mv 2 s /R with v s being the speed on the surface.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric Surface Momentum and Centripetal Force for a Particle on a Curved Surface

Shikakhwa

2018

Commun. Theor. Phys.

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The Hermitian surface momentum operator for a particle confined to a 2D curved surface spanned by orthogonal coordinates and embedded in 3D space is expressed as a symmetric expression in derivatives with respect to the surface coordinates and so is manifestly along the surface. This is an alternative form to the one reported in the literature and usually named geometric momentum, which has a term proportional to the mean curvature along the direction normal to the surface, and so "apparently" not along the surface. The symmetric form of the momentum is the sum of two symmetric Hermitian operators along the two orthogonal directions defined by the surface coordinates. The centripetal force operator for a particle on the surface of a cylinder and a sphere is calculated by taking the time derivative of the momentum and is seen to be a symmetrization of the well-known classical expressions.

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Section: Introductionsupporting

confidence: 61%

Section: The Symmetric Form Of the Hermitian Surface Momentummentioning

confidence: 89%

Section: The Symmetric Form Of the Hermitian Surface Momentummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetric Surface Momentum and Centripetal Force for a Particle on a Curved Surface

Shikakhwa

2018

Commun. Theor. Phys.

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Geometric Potential and Dirac Quantization

Lian

Liu

2018

Annalen der Physik

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A fundamental problem regarding the Dirac quantization of a free particle on an N − 1 (N ≥ 2) curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature-induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin-layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator-ordering-free section is identified and is then found sufficient to lead to the expected form of geometric potential.

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Can Dirac quantization of constrained systems be fulfilled within the intrinsic geometry?

Xun

Liu

2014

Annals of Physics

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Geometric Momentum and a Probe of Embedding Effects

Cited by 22 publications

References 42 publications

Symmetric Surface Momentum and Centripetal Force for a Particle on a Curved Surface

Symmetric Surface Momentum and Centripetal Force for a Particle on a Curved Surface

Geometric Potential and Dirac Quantization

Can Dirac quantization of constrained systems be fulfilled within the intrinsic geometry?

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