THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`

2021 ◽  
Vol 10 (9) ◽  
pp. 3253-3262
Author(s):  
H. Umair ◽  
H. Zainuddin ◽  
K.T. Chan ◽  
Sh.K. Said Husein
Keyword(s):  
Uncertainty Principle ◽  
Symplectic Form ◽  
Uncertainty Relation ◽  
Riemannian Metric ◽  
Hamiltonian Flow ◽  
Geometric Uncertainty ◽  
Projective Hilbert Space ◽  
Space Dynamics ◽  
Geometric Quantum Mechanics ◽  
Complex Projective

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.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

2021 ◽  
Vol 10 (9) ◽  
pp. 3241-3251
Author(s):  
H. Umair ◽  
H. Zainuddin ◽  
K.T. Chan ◽  
Sh.K. Said Husein
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Uncertainty Relation ◽  
Dynamical Behavior ◽  
Dynamical Evolution ◽  
Riemannian Metric ◽  
Hamiltonian Flow ◽  
Space Dynamics ◽  
Geometric Quantum Mechanics ◽  
Complex Projective

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.

scholarly journals The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

2016 ◽  
Vol 08 (03) ◽  
pp. 545-570 ◽  
Author(s):  
Luca Asselle ◽  
Gabriele Benedetti
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Critical Value ◽  
Closed Manifold ◽  
Hamiltonian Flow ◽  
Cotangent Bundles ◽  
Almost All

Let [Formula: see text] be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian [Formula: see text] and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if [Formula: see text] is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of [Formula: see text].

scholarly journals CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES

2010 ◽  
Vol 07 (03) ◽  
pp. 485-503 ◽  
Author(s):  
P. ANIELLO ◽  
J. CLEMENTE-GALLARDO ◽  
G. MARMO ◽  
G. F. VOLKERT
Keyword(s):  
Hilbert Space ◽  
Symplectic Geometry ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Inner Product ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Separable States ◽  
Pure States ◽  
Complex Projective

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

scholarly journals The super-Sasaki metric on the antitangent bundle

2020 ◽  
Vol 17 (08) ◽  
pp. 2050122
Author(s):  
Andrew James Bruce
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Symplectic Form ◽  
Riemannian Metric ◽  
Riemannian Structure ◽  
Sasaki Metric

We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalization of Sasaki’s construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.

scholarly journals Geometric uncertainty relation for quantum ensembles

2015 ◽  
Vol 90 (2) ◽  
pp. 025102 ◽  
Author(s):  
Hoshang Heydari ◽  
Ole Andersson
Keyword(s):  
Uncertainty Relation ◽  
Geometric Uncertainty ◽  
Quantum Ensembles

Is the Uncertainty Relation at the Root of all Mutual Exclusiveness?

1997 ◽  
Vol 52 (5) ◽  
pp. 398-402 ◽  
Author(s):  
D. Sen ◽  
A. N. Basu ◽  
S. Sengupta
Keyword(s):  
Interference Pattern ◽  
Uncertainty Principle ◽  
Uncertainty Relation ◽  
Quantum Mechanical ◽  
Complementarity Principle ◽  
Conventional Analysis ◽  
Double Slit Experiment ◽  
Slit Experiment ◽  
Double Slit

Abstract It is argued that two distinct types of complementarity are implied in Bohr's complementarity principle. While in the case of complementary variables it is the quantum mechanical uncertainty relation which is at work, the collapse hypothesis ensures this exclusiveness in the so-called wave-particle complementarity experiments. In particular it is shown that the conventional analysis of the double slit experiment which invokes the uncertainty principle to explain the absence of the simultaneous knowledge of the which-slit information and the interference pattern is incorrect and implies consequences that are quantum mechanically inconsistent.

scholarly journals Geometric uncertainty relation for mixed quantum states

2014 ◽  
Vol 55 (4) ◽  
pp. 042110 ◽  
Author(s):  
Ole Andersson ◽  
Hoshang Heydari
Keyword(s):  
Uncertainty Relation ◽  
Quantum States ◽  
Geometric Uncertainty

UNCERTAINTY PRINCIPLE, SQUEEZING, AND QUANTUM GROUPS

1992 ◽  
Vol 07 (40) ◽  
pp. 3759-3764 ◽  
Author(s):  
A.K. RAJAGOPAL ◽  
VIRENDRA GUPTA
Keyword(s):  
Quantum Mechanics ◽  
Coherent State ◽  
Quantum Groups ◽  
Uncertainty Principle ◽  
Unitary Transformation ◽  
Uncertainty Relation ◽  
Heisenberg Uncertainty Relation ◽  
Heisenberg Uncertainty ◽  
Complete Form

It is shown that the complete form of the Heisenberg Uncertainty Relation (HUR) must be employed in introducing the concepts of squeezing and coherent state in q-quantum mechanics. An important feature of this form of the HUR is that it is invariant under unitary transformation of the operators appearing in it and consequences of this are pointed out.

UNCERTAINTY PRINCIPLE AND THE QUANTUM FLUCTUATIONS OF THE SCHWARZSCHILD LIGHT CONES

1986 ◽  
Vol 01 (02) ◽  
pp. 491-498 ◽  
Author(s):  
T. PADMANABHAN ◽  
T.R. SESHADRI ◽  
T.P. SINGH
Keyword(s):  
Gravitational Field ◽  
Event Horizon ◽  
Uncertainty Principle ◽  
Light Cone ◽  
Uncertainty Relation ◽  
Quantum Fluctuations ◽  
Point Mass ◽  
Conjugate Momentum

We consider the gravitational field of a point mass and show that the application of the uncertainty principle leads to (i) an uncertainty relation for the metric and its conjugate momentum and (ii) finite fluctuations of the light-cone at the event horizon.

scholarly journals SPACE–TIME UNCERTAINTY PRINCIPLE FROM BREAKDOWN OF TOPOLOGICAL SYMMETRY

1998 ◽  
Vol 13 (03) ◽  
pp. 203-209 ◽  
Author(s):  
ICHIRO ODA
Keyword(s):  
Uncertainty Principle ◽  
Uncertainty Relation ◽  
Space Time ◽  
Topological Quantum Field Theory ◽  
Time Interval ◽  
Quantum Field ◽  
Higher Symmetry ◽  
Time Uncertainty ◽  
Topological Quantum ◽  
Large N

Starting from topological quantum field theory, we derive space–time uncertainty relation with respect to the time interval and the spatial length proposed by Yoneya through breakdown of topological symmetry in the large-N matrix model. This work suggests that the topological symmetry might be an underlying higher symmetry behind the space–time uncertainty principle of string theory.

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