Classical aspect of uncertainty principle for spin angular momentum in geometric quantum mechanics

Quantum mechanics is one of two foundational parts of modern physics. Along with relativity, quantum mechanics plays a central roles in explaining the nature and behavior of matter on the microscopic level. It is regarded as most successful theory ever developed in history of physics. However it is difficult to make a smooth connection between classical mechanics and quantum mechanics since classical mechanics is based on geometry and some of the systems are non-linear whereas quantum mechanics is intrinsically algebraic and linear. The fact that classical mechanics, general relativity and others are highly geometrical inspired some physicists to cast quantum mechanics in geometrical language in order to better understand the quantum-classical transition. Within this framework the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are real valued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this research, the properties of spin 1 2 , spin 1 and spin 32 particles in geometric quantum mechanics framework have been studied. Generally the Robertson-Schrodinger uncertainty principle for these systems has been demonstrated varies along any Hamiltonian flows. This work was done by calculating the evolution of symplectic area and component of Riemannian metric under the flows. Besides, the correspondence between Poisson bracket and commutator for these systems was showed by explicitly computed the value of commutator of spin operators and compared it with the Poisson bracket of the corresponding classical observables. This study was extended by comparing the Casimir operator and its classical counterpart. The results showed that there exist correspondence between classical and quantum Casimir operator at least for the case of spin 12 . This research might be a good step toward inserting the aspect of symplectic topology such as non-squeezing theorem and clearly showed the limit of classical notion to describe the purely quantum concept.

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