Quantum Mechanics in Riemannian Manifold. II

Ogawa, Naohisa; Fujii, Kanji; Chepilko, N.M.; Kobushkin, A. P.

doi:10.1143/ptp.85.1189

Cited by 30 publications

(47 citation statements)

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“…4), are exactly 3 as predicted by Eq. (9). Notice that they are all converging to the flat case value of π 2 for very small curvature.…”

Section: B Two Dimensionsmentioning

confidence: 74%

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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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“…4), are exactly 3 as predicted by Eq. (9). Notice that they are all converging to the flat case value of π 2 for very small curvature.…”

Section: B Two Dimensionsmentioning

confidence: 74%

“…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%

Polymers in Curved Boxes

Yaman¹,

Pincus²,

Solis³

et al. 1997

Macromolecules

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“…It must be mentioned that Dirac himself had never assumed so except when the particle moves in flat space where we should directly quantize the Poisson brackets, and he was clearly aware of the operator‐ordering difficulties which should be carefully got over . If taking the straightforward definition of quantum condition for different components of the momentum,

[p_{i}, p_{j}]

, we encounter a disturbing operator‐ordering problem in

O \{(n_{j} n_{i, k} - n_{i} n_{j, k}) p_{k}\}

. Much more annoying operator‐ordering problem appears in

O \{boldn false(boldp \cdot \nabla boldn \cdot boldp false)\}

if one attempts to construct a quantum condition for

[p, H]

…”

Section: Dirac Brackets and Quantization Conditionsmentioning

confidence: 99%

“…It is curious that no attempt is successful for even simplest two‐dimensional curved surface Σ 2 embedded in R 3 . Some results are contradictory with each other . We revisited all these attempts, and concluded that the canonical quantization together with Schrödinger–Podolsky–DeWitt approach of Hamiltonian operator construction was dubious, for the kinetic energy in it takes some presumed forms of distributing positions and momenta in the Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

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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“…60-66 and references therein). This problem is often referred as the quantization in Riemannian or curved space by the analogy to the differential geometry of surfaces [62]. Because generalized momentas are differential operators, it is more difficult to obtain the quantum Hamiltonian in the presence of constraints compared to the classical case (see the end of Section II.B).…”

Section: B Constrained Quantizationmentioning

confidence: 99%