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

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.

differential geometry
uncertainty principle
geometric quantum mechanics
quantum dynamics
Hamiltonian mechanics
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