On bounded partition dimension of different families of convex polytopes with pendant edges

Let ψ=(V,E) be a simple connected graph. The distance between ρ1,ρ2∈V(ψ) is the length of a shortest path between ρ1 and ρ2. Let Γ={Γ1,Γ2,…,Γj} be an ordered partition of the vertices of ψ . Let ρ1∈V(ψ) , and r(ρ1|Γ)={d(ρ1,Γ1),d(ρ1,Γ2),…,d(ρ1,Γj)} be a j -tuple. If the representation r(ρ1|Γ) of every ρ1∈V(ψ) w.r.t. Γ is unique then Γ is the resolving partition set of vertices of ψ . The minimum value of j in the resolving partition set is known as partition dimension and written as pd(ψ). The problem of computing exact and constant values of partition dimension is hard so one can compute bound for the partition dimension of a general family of graph. In this paper, we studied partition dimension of the some families of convex polytopes with pendant edge such as RPn , Dpn and Qpn and proved that these graphs have bounded partition dimension

The dynamical evolution of geometric uncertainty principle for spin 1/2 system

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow

The generalized geometric uncertainty principle for spin 1/2 system

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 Schrödinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schrödinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schrödinger uncertainty principle for spin ½ system has been constructed by considering the operators corresponding to arbitrary direction

The evolution of geometric Robertson–Schrodinger 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 Schr¨odinger 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–Schr¨odinger 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–Schr¨odinger 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