Quantum mechanics of a particle on a torus knot: Curvature and torsion effects

Biswas, Dripto; Ghosh, Subir

doi:10.1209/0295-5075/132/10004

Cited by 9 publications

(5 citation statements)

References 19 publications

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“…These results are correct only in a flat space. If the curve is embedded into the space of nonzero curvature, the above theorems must be amended [134]. Moreover, they must be amended if the curve is replaced by a beam of, say, the circular cross-section α.…”

Section: Differential Geometry Of Elastic Curves and Kirchhoff's Elas...mentioning

confidence: 99%

Maxwell-Dirac Isomorphism Revisited: From Foundations of Quantum Mechanics to Geometrodynamics and Cosmology

Kholodenko

2023

Universe

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Although electrons (fermions)and photons (bosons) produce the same interference patterns in the two-slit experiments, known in optics for photons since the 17th Century, the description of these patterns for electrons and photons thus far was markedly different. Photons are spin one, relativistic and massless particles while electrons are spin half massive particles producing the same interference patterns irrespective to their speed. Experiments with other massive particles demonstrate the same kind of interference patterns. In spite of these differences, in the early 1930s of the 20th Century, the isomorphism between the source-free Maxwell and Dirac equations was established. In this work, we were permitted replace the Born probabilistic interpretation of quantum mechanics with the optical. In 1925, Rainich combined source-free Maxwell equations with Einstein’s equations for gravity. His results were rediscovered in the late 1950s by Misner and Wheeler, who introduced the word "geometrodynamics” as a description of the unified field theory of gravity and electromagnetism. An absence of sources remained a problem in this unified theory until Ranada’s work of the late 1980s. However, his results required the existence of null electromagnetic fields. These were absent in Rainich–Misner–Wheeler’s geometrodynamics. They were added to it in the 1960s by Geroch. Ranada’s solutions of source-free Maxwell’s equations came out as knots and links. In this work, we establish that, due to their topology, these knots/links acquire masses and charges. They live on the Dupin cyclides—the invariants of Lie sphere geometry. Symmetries of Minkowski space-time also belong to this geometry. Using these symmetries, Varlamov recently demonstrated group-theoretically that the experimentally known mass spectrum for all mesons and baryons is obtainable with one formula, containing electron mass as an input. In this work, using some facts from polymer physics and differential geometry, a new proof of the knotty nature of the electron is established. The obtained result perfectly blends with the description of a rotating and charged black hole.

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Section: Differential Geometry Of Elastic Curves and Kirchhoff's Elas...mentioning

confidence: 99%

Maxwell-Dirac Isomorphism Revisited: From Foundations of Quantum Mechanics to Geometrodynamics and Cosmology

Kholodenko

2023

Universe

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“…As we shall see, we focus on the investigation of the thermal quantities in a numerical way. In past years, the analysis of quantum mechanics within the context of confined systems has received special attention due to its variety of applications: from theoretical through experimental viewpoints [44][45][46][47][48][49][50][51][52][53]. In particular, it was Jones [54] who pioneer studied knot invariants to derive the correlation between pure mathematics and the physical world.…”

Section: A Thermodynamic Approachmentioning

confidence: 99%

“…On the other hand, in Ref. [51], the respective spectral energy for the torus knot was first calculated. Based on it, we intend to examine how particles of different spins behave in this configuration.…”

Section: A Thermodynamic Approachmentioning

confidence: 99%

Quantum gases on a torus

Filho¹,

Reis²,

Ghosh³

2022

Preprint

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This manuscript is aimed at studying the thermodynamic properties of quantum gases confined to a torus. To do that, we consider noninteracting gases within the grand canonical ensemble formalism. In this context, fermoins and bosons are taken into account and the calculations are properly provided in both analytical and numerical manners. In particular, the system turns out to be sensitive to the topological parameter under consideration: the winding number. Furthermore, we also derive a model in order to take into account interacting quantum gases. To corroborate our results, we implement such a method for two different scenarios: a ring and a torus.

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“…In this context, several examples of exactly solvable models, like the harmonic oscillator [2], the two-body problem with non-central forces [3], the modified ring-shaped oscillator potential [4], and a model involving a class of hyperbolic potential well [5]. Because of its fundamental feature, it is possible to think about the Schrödinger equation in the most diverse contexts, describing low-dimensional electron gases [6], problems with anisotropic mass [7], as well as the presence of curvature and torsion in the spacetime [8]. Concerning the Schrödinger equation in a curved space, a relevant type of problem consists of studying the presence of topological defects [9].…”

Section: Introductionmentioning

confidence: 99%

Approximate Analytical Solutions of the Schrödinger Equation with Hulthén Potential in the Global Monopole Spacetime

et al. 2023

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In this paper, we studied the nonrelativistic quantum mechanics of an electron in a spacetime containing a topological defect. We also considered that the electron is influenced by the Hulthén potential. In particular, we dealt with the Schrödinger equation in the presence of a global monopole. We obtained approximate solutions for the problem, determined the scattering phase shift and the S-matrix, and analyzed bound states.

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Quantum mechanics of a particle on a torus knot: Curvature and torsion effects

Cited by 9 publications

References 19 publications

Maxwell-Dirac Isomorphism Revisited: From Foundations of Quantum Mechanics to Geometrodynamics and Cosmology

Maxwell-Dirac Isomorphism Revisited: From Foundations of Quantum Mechanics to Geometrodynamics and Cosmology

Quantum gases on a torus

Approximate Analytical Solutions of the Schrödinger Equation with Hulthén Potential in the Global Monopole Spacetime

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