Schrödinger Equation: Symmetries and Nanoscale Dynamics

Andrzej Jakubiec, Jerzy Kijowski

Ndenga, Barack

The coexistence of quantum coherence, nonlinear fluid dynamics, and biomolecular π-fields within biological and nanoscale systems requires a unified mathematical description. Classical hydrodynamics alone fails to capture coherence propagation, while the linear Schrödinger equation cannot model dissipation, viscosity, or turbulence. Here, I introduce the Schrödinger–Navier–Stokes–π Unified Computational Framework, a novel equation set combining: the quantum phase field of the Schrödinger equation, the viscous and nonlinear transport of Navier–Stokes dynamics, the curvature-driven quantum π-potential, and a hybrid Hamiltonian–dissipative evolution capturing both coherence and irreversible flow. This framework predicts quantum-fluid transitions, π-induced tunneling acceleration, coherence-enhanced mixing, nanoscale turbulence, and dynamic switching between classical and quantum transport regimes. It provides a general model applicable to biology, nanotechnology, photonics, and quantum materials. Keywords : Quantum Hydrodynamics Navier–Stokes Schrödinger Equation Unified Computational Framework π-field Dynamics Quantum Fluid Mechanics Hybrid Quantum-Classical Transport Coherence-Based Models Quantum Biology Nanoscale Transport Bio-Quantum Modeling Quantum Potential Quantum Fluid Engineering Biophysical Modeling Nonlinear Dynamics Computational Physics High-Order Numerical Methods Dissipative Quantum Systems Biomolecular Channels Quantum-Inspired Computing

V. M. Somsikov

Extension of the Schrodinger equation is submitted by removing its limitations appearing due to the limitations of the formalism of Hamilton, based on which this equation was obtained. For this purpose the problems of quantum mechanics arising from the limitations of classical mechanics are discussed. These limitations, in particular, preclude the use of the Schrodinger equation to describe the time symmetry violation. The extension of the Schrodinger equation is realized based on the principle of duality symmetry. According to this principle the dynamics of the systems is determined by the symmetry of the system and by the symmetry of the space. The extension of the Schrodinger equation was obtained from the dual expression of energy, represented in operator form. For this purpose the independent micro - and macro-variables that determine respectively the dynamics of quantum particle system relative to its center of mass and the movement of the center of mass in space are used. The solution of the extended Schrodinger equation for the system near equilibrium is submitted. The main advantage of the extended Schrodinger equation is that it is applicable to describe the interaction and evolution of quantum systems in inhomogeneous field of external forces.

Leon Bess

Vyacheslav Michailovich Somsikov, A.A. Idrissov, V. I. Kapytin +2 more

Broadening the Schrödinger equation by removing the limitations imposed by the formalism of Hamilton, based on which this equation received is submitted. These limitations pre- clude the use of the Schrödinger equation to describe the dynamical symmetry violation. The expan- sion of the Schrödinger equation is realized based on the principle of duality symmetry. According to this principle the dynamic of the systems determined by the symmetry of the system and space. The extended Schrödinger equation was obtained from the dual expression of energy, represented in operator form. The independent micro and macro variables that determine respectively the dynamics of micro-particle quantum system relative to its center of mass and the movement of the center of mass in space are used. The problems of quantum mechanics, due to the structural change of micro particles and the internal energy of the strong interactions are discussed. Keywords: Schrödinger equation, the principle of duality symmetries, dual coordinate system.