The Deterministic and Geometric Equation of the Total Particle Lifetime Illusion in Quantum Phase Transitions

We consider an {\em elastic} helical nanoribbon that can take on various conformations, and study the effect of placing a quantum particle on its curved surface. Using a modified Canham-Helfrich model for the elastic energy, we write down the local elastic potential for the ribbon in terms of its bending rigidity, mean curvature $M$ and Gaussian curvature $K$ . The Schrödinger equation of a particle confined to a {\em rigid} curved surface is found using da Costa's formulation. It has a purely quantum geometric potential which depends on $M$ and $K$ . The Schrödinger equation of a particle on an {\em elastic } curved surface will therefore have a total potential comprising quantum and elastic potentials. We compute $M$ and $K$ for a helical ribbon and derive the total potential which depends on the conformation and is thus geometric in nature. Defining a dimensionless quantity $R_H$ , we study the behavior of the total geometric potential as $R_H$ is varied. In the absence of an electron, the elastic potential is positive and has a single positive maximum for all conformations. Further, a binormal helical ribbon conformation has the lowest potential, while the normal ribbon has the highest, with those of the intermediate ribbons lying in between these. Intriguingly, when a quantum particle is placed on the elastic ribbon, above a certain critical value of $R_H$ , the presence of the quantum geometric potential {\em reverses} this order. But localized states for the particle are not supported. Only above a second critical value of $R_H$ , localized states appear for all conformations. The injection of an electron on {\it any} given conformation of the elastic ribbon will induce a conformational transition to the normal ribbon conformation.We consider an {\em elastic} helical nanoribbon that can take on various conformations, and study the effect of placing a quantum particle on its curved surface. Using a modified Canham-Helfrich model for the elastic energy, we write down the local elastic potential for the ribbon in terms of its bending rigidity, mean curvature $M$ and Gaussian curvature $K$. The Schrödinger equation of a particle confined to a {\em rigid} curved surface is found using da Costa's formulation. It has a purely quantum geometric potential which depends on $M$ and $K$. The Schrödinger equation of a particle on an {\em elastic } curved surface will therefore have a total potential comprising quantum and elastic potentials. We compute $M$ and $K$ for a helical ribbon and derive the total potential which depends on the conformation and is thus geometric in nature. Defining a dimensionless quantity $R_H$, we study the behavior of the total geometric potential as $R_H$ is varied. In the absence of an electron, the elastic potential is positive and has a single positive maximum for all conformations. Further, a binormal helical ribbon conformation has the lowest potential, while the normal ribbon has the highest, with those of the intermediate ribbons lying in between these. Intriguingly, when a quantum particle is placed on the elastic ribbon, above a certain critical value of $R_H$, the presence of the quantum geometric potential {\em reverses} this order. But localized states for the particle are not supported. Only above a second critical value of $R_H$, localized states appear for all conformations. The injection of an electron on {\it any} given conformation of the elastic ribbon will induce a conformational transition to the normal ribbon conformation.

quant-pharXiv:2607.29623Aug 3, 20260 citations

Geometric phase of quantum wave function and singularities of Bohm dynamics in a one-dimensional system

Omar Morandi

Abstract The interpretation of the quantum mechanics proposed by de Broglie and Bohm postulates that the time evolution of the position and the momentum of a quantum particle can be described by a trajectory in the phase-space. The evolution equation coincides with the classical one except for the presence of a nonlinear correction to the total energy of the particle denoted by Bohm potential. The particle momentum is associated to the derivative of the phase of the quantum wave function. The phase of a quantum wave function ceases to be globally well defined in the presence of zeros (nodes) and the Bohm potential becomes singular. We develop a geometrical interpretation of the Bohm dynamics based on the Ehresmann theory of the fiber bundles and we express the number of total rotations of the quantum phase around a node in terms of holonomy maps.Abstract The interpretation of the quantum mechanics proposed by de Broglie and Bohm postulates that the time evolution of the position and the momentum of a quantum particle can be described by a trajectory in the phase-space. The evolution equation coincides with the classical one except for the presence of a nonlinear correction to the total energy of the particle denoted by Bohm potential. The particle momentum is associated to the derivative of the phase of the quantum wave function. The phase of a quantum wave function ceases to be globally well defined in the presence of zeros (nodes) and the Bohm potential becomes singular. We develop a geometrical interpretation of the Bohm dynamics based on the Ehresmann theory of the fiber bundles and we express the number of total rotations of the quantum phase around a node in terms of holonomy maps.

Journal of Physics A Mathematical and TheoreticalNov 30, 20221 citation

Explaining the Geometric Phase and Adiabatic Evolution of a Quantum System

Ijaz Durrani

Jan 1, 20260 citations

The $\Omega$ GLR Translation Matrix

Marco Lindenbeck

Title: The $\Omega$ GLR Translation Matrix - Translating Standard Model Mathematics into Substrate Logistics Author: Marco Lindenbeck Description: The Standard Model of particle physics and Quantum Field Theory (QFT) provide exceptionally precise mathematical models of reality, yet they rely on the philosophical assumption of a continuous, probabilistic, and often paradoxical vacuum. This manuscript serves as the formal Translation Matrix for the Omega Grounded Light Reality ( $\Omega$ GLR) framework. It systematically maps the core mathematical constructs of classical and quantum mechanics—including the Lagrangian, the Hamiltonian, the Schrödinger Equation, Feynman Diagrams, commutator algebra, and Bell's Inequality—directly onto a deterministic, discrete topological hardware substrate ( $\kappa=50$ ). By reinterpreting these established equations through the lens of Substrate Logistics, the framework permanently resolves the mysticism of quantum probability, wave-function collapse, and virtual particles. The Principle of Least Action is formally redefined as a Topological Pathfinding Algorithm minimizing spatial friction. The Schrödinger wave function is translated into classical topological acoustics within a 12-face dodecahedral metric cell. Furthermore, Bell's non-locality is mechanically resolved by defining quantum entanglement as strict 1-node backend topological adjacency, bypassing the rendered illusion of macroscopic Euclidean distance. The underlying mathematics of the Standard Model are preserved flawlessly, but rigidly recontextualized as the deterministic software operations of a finite-capacity, geometric architecture.Title: The $\Omega$GLR Translation Matrix - Translating Standard Model Mathematics into Substrate Logistics Author: Marco Lindenbeck Description: The Standard Model of particle physics and Quantum Field Theory (QFT) provide exceptionally precise mathematical models of reality, yet they rely on the philosophical assumption of a continuous, probabilistic, and often paradoxical vacuum. This manuscript serves as the formal Translation Matrix for the Omega Grounded Light Reality ($\Omega$GLR) framework. It systematically maps the core mathematical constructs of classical and quantum mechanics—including the Lagrangian, the Hamiltonian, the Schrödinger Equation, Feynman Diagrams, commutator algebra, and Bell's Inequality—directly onto a deterministic, discrete topological hardware substrate ($\kappa=50$). By reinterpreting these established equations through the lens of Substrate Logistics, the framework permanently resolves the mysticism of quantum probability, wave-function collapse, and virtual particles. The Principle of Least Action is formally redefined as a Topological Pathfinding Algorithm minimizing spatial friction. The Schrödinger wave function is translated into classical topological acoustics within a 12-face dodecahedral metric cell. Furthermore, Bell's non-locality is mechanically resolved by defining quantum entanglement as strict 1-node backend topological adjacency, bypassing the rendered illusion of macroscopic Euclidean distance. The underlying mathematics of the Standard Model are preserved flawlessly, but rigidly recontextualized as the deterministic software operations of a finite-capacity, geometric architecture.

Jul 1, 20260 citations

A microscopic approach to phase transitions in quantum systems

Pietro Gianinetti, Alberto Parola

arXiv:cond-mat/0002033Physics Letters AApr 1, 20002 citations

M

and Gaussian curvature

K

. The Schrödinger equation of a particle confined to a {\em rigid} curved surface is found using da Costa's formulation. It has a purely quantum geometric potential which depends on

M

and

K

. The Schrödinger equation of a particle on an {\em elastic } curved surface will therefore have a total potential comprising quantum and elastic potentials. We compute

M

and

K

for a helical ribbon and derive the total potential which depends on the conformation and is thus geometric in nature. Defining a dimensionless quantity

R_H

, we study the behavior of the total geometric potential as

R_H

is varied. In the absence of an electron, the elastic potential is positive and has a single positive maximum for all conformations. Further, a binormal helical ribbon conformation has the lowest potential, while the normal ribbon has the highest, with those of the intermediate ribbons lying in between these. Intriguingly, when a quantum particle is placed on the elastic ribbon, above a certain critical value of

R_H

, the presence of the quantum geometric potential {\em reverses} this order. But localized states for the particle are not supported. Only above a second critical value of

R_H

, localized states appear for all conformations. The injection of an electron on {\it any} given conformation of the elastic ribbon will induce a conformational transition to the normal ribbon conformation.

Title: The

\Omega

GLR Translation Matrix - Translating Standard Model Mathematics into Substrate Logistics Author: Marco Lindenbeck Description: The Standard Model of particle physics and Quantum Field Theory (QFT) provide exceptionally precise mathematical models of reality, yet they rely on the philosophical assumption of a continuous, probabilistic, and often paradoxical vacuum. This manuscript serves as the formal Translation Matrix for the Omega Grounded Light Reality (

\Omega

GLR) framework. It systematically maps the core mathematical constructs of classical and quantum mechanics—including the Lagrangian, the Hamiltonian, the Schrödinger Equation, Feynman Diagrams, commutator algebra, and Bell's Inequality—directly onto a deterministic, discrete topological hardware substrate (

\kappa=50

). By reinterpreting these established equations through the lens of Substrate Logistics, the framework permanently resolves the mysticism of quantum probability, wave-function collapse, and virtual particles. The Principle of Least Action is formally redefined as a Topological Pathfinding Algorithm minimizing spatial friction. The Schrödinger wave function is translated into classical topological acoustics within a 12-face dodecahedral metric cell. Furthermore, Bell's non-locality is mechanically resolved by defining quantum entanglement as strict 1-node backend topological adjacency, bypassing the rendered illusion of macroscopic Euclidean distance. The underlying mathematics of the Standard Model are preserved flawlessly, but rigidly recontextualized as the deterministic software operations of a finite-capacity, geometric architecture.