Schrödinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space

Homma, Teiichi; Inamoto, Tadashi; Miyazaki, Tadashi

doi:10.1103/physrevd.42.2049

Cited by 37 publications

(78 citation statements)

References 5 publications

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“…There are also studies of the contribution of torsion to the bound state energies, [3], [4]. Until now the results for higher dimensional confinement are restricted to the perturbative regime, [5], [6], [7], [8], [9], [3], [10].…”

Section: Introductionmentioning

confidence: 99%

“…The correct quantization procedure for quantum particles confined to manifolds embedded in higher dimensional spaces turns out to be non-trivial, [5]. Studies, [6] - [12], which are perturbative in the curvature of the manifold, conclude that confinement to a curved manifold results in an effective potential, which depends both on the manifold to which the particle is confined and the particular way this manifold is embedded in the higher dimensional space, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Polymers in Curved Boxes

Yaman¹,

Pincus²,

Solis³

et al. 1997

Macromolecules

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We apply results derived in other contexts for the spectrum of the Laplace operator in curved geometries to the study of an ideal polymer chain confined to a spherical annulus in arbitrary space dimension D and conclude that the free energy compared to its value for an uncurved box of the same thickness and volume, is lower when D < 3, stays the same when D = 3, and is higher when D > 3. Thus confining an ideal polymer chain to a cylindrical shell, lowers the effective bending elasticity of the walls, and might induce spontaneous symmetry breaking, i.e. bending. (Actually, the above mentioned results show that any shell in D = 3 induces this effect, except for a spherical shell). We compute the contribution of this effect to the bending rigidities in the Helfrich free energy expression.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polymers in Curved Boxes

Yaman¹,

Pincus²,

Solis³

et al. 1997

Macromolecules

View full text Add to dashboard Cite

show abstract

“…(3) is under dispute. For instance, Hamma et al [4] and Ikegami et al [5] showed that from the constraint equation f (r) = 0 we could not build up a satisfactory theory and we must start from another constraint equation d f (r)/dt = 0, but Weinberg [6] thought that Eq. (3) should be as true as it is in classical mechanics.…”

Section: Introductionmentioning

confidence: 99%

“…The motion of a particle on a curved hypersurface is an exactly solvable model to examine various problems such as higher-dimensional gravity [1], the dark energy/matter problem [2], the quantization of constrained motions for a nonrelativistic particle [3][4][5][6][7] and for a relativistic fermion [8,9], and many curvature-induced effects in lower-dimensional systems and nanostructures [10][11][12][13][14][15], etc. We are familiar with both the geodesic equation from the intrinsically curved surface and the equation of motion from the extrinsically Euclidean space [5,6], but no relationship in between has been seriously explored.…”

Section: Introductionmentioning

confidence: 99%

The centripetal force law and the equation of motion for a particle on a curved hypersurface

Lian

Liu

2016

Eur. Phys. J. C

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It is pointed out that the current form of the extrinsic equation of motion for a particle constrained to remain on a hypersurface is in fact a half-finished version; for it is established without regard to the fact that the particle can never depart from the geodesics on the surface. Once this fact is taken into consideration, the equation takes the same form as that for the centripetal force law, provided that the symbols are re-interpreted so that the law is applicable for higher dimensions. The controversial issue of constructing operator forms of these equations is addressed, and our studies show the quantization of constrained system based on the extrinsic equation of motion is preferable.

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“…Historically, at the level of the non-relativistic quantum mechanics, some problems were solved within this frame- * E-mail: meradm@gmail.com work for example the Schrodinger equation particle in the presence of constant gravity [1], path integrals for a particle in curved space [2], the nonrelativistic particle constrained on a hypersurface in a curved space [3].…”

Section: Introductionmentioning

confidence: 99%