Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.

Advances in Mathematics Scientific JournalSep 28, 20210 citations

The evolution of geometric Robertson–Schrödinger uncertainty principle for spin 1 system

Umair Abdul Halim, Hishamuddin B Zainuddin, Kar Tim Chan +1 more

Geometric Quantum Mechanics is a mathematical framework that shows how quantum theory may be expressed in terms of Hamiltonian phase-space dynamics. The states are points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is specified by the Schrödinger equation in this framework. The quest to express the uncertainty principle in geometrical language has recently become the focus of significant research in geometric quantum mechanics. One has demonstrated that the Robertson–Schrödinger uncertainty principle, which is a stronger version of the uncertainty relation, can be defined in terms of symplectic form and Riemannian metric. On the basis of this formulation, we study the dynamical behavior of the uncertainty relation for the spin 1 system in this work. We show that under Hamiltonian flow, the Robertson–Schrödinger uncertainty principles are not invariant. This is because, unlike the symplectic area, the Riemannian metric is not invariant under Hamiltonian flow throughout the evolution process.Geometric Quantum Mechanics is a mathematical framework that shows how quantum theory may be expressed in terms of Hamiltonian phase-space dynamics. The states are points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is specified by the Schrödinger equation in this framework. The quest to express the uncertainty principle in geometrical language has recently become the focus of significant research in geometric quantum mechanics. One has demonstrated that the Robertson–Schrödinger uncertainty principle, which is a stronger version of the uncertainty relation, can be defined in terms of symplectic form and Riemannian metric. On the basis of this formulation, we study the dynamical behavior of the uncertainty relation for the spin 1 system in this work. We show that under Hamiltonian flow, the Robertson–Schrödinger uncertainty principles are not invariant. This is because, unlike the symplectic area, the Riemannian metric is not invariant under Hamiltonian flow throughout the evolution process.

Mathematical Modeling and ComputingJan 1, 20220 citations

Generalized uncertainty relations and efficient measurements in quantum systems

V. P. Belavkin

arXiv:quant-ph/0412030Theoretical and Mathematical PhysicsMar 1, 197658 citations

Fractal Phase-Space Uncertainty: A Logical-Spectral Reformulation of the Heisenberg Principle

Enrique Vidal Silvente

We develop a fractal-spectral reformulation of the Heisenberg uncertainty principle within a nonlinear phase-space framework motivated by the Quantum Fractal-Logical Unified Field architecture. Instead of working on the standard Hilbert space L 2 (R) with Lebesgue measure and linear position/momentum operators, we consider a fractal Hilbert space H F = L 2 (F, µ H) built on a Cantor-type metric-measure substrate with Hausdorff measure µ H and a self-adjoint fractal Hamiltonian. Within this setting we define effective position and momentum observables (X F , P F) adapted to the fractal geometry and derive a generalized uncertainty relationWe develop a fractal-spectral reformulation of the Heisenberg uncertainty principle within a nonlinear phase-space framework motivated by the Quantum Fractal-Logical Unified Field architecture. Instead of working on the standard Hilbert space L 2 (R) with Lebesgue measure and linear position/momentum operators, we consider a fractal Hilbert space H F = L 2 (F, µ H) built on a Cantor-type metric-measure substrate with Hausdorff measure µ H and a self-adjoint fractal Hamiltonian. Within this setting we define effective position and momentum observables (X F , P F) adapted to the fractal geometry and derive a generalized uncertainty relation

Mar 18, 20260 citations

\frac{1}{2}

system has been constructed by considering the operators corresponding to arbitrary direction.