Generalized Uncertainty Relations in the Non-commutative Plane

Won Sang Chung

Pritam Chattopadhyay, Ayan Mitra, Goutam Paul

Krzysztof Urbanowski

We analyze the Schwarz inequality and its generalizations, as well as inequalities resulting from the Jensen inequality. They are used in quantum theory to derive the Heisenberg-Robertson (HR) and Schroedinger-Robertson (SR) uncertainty relation for two non-commuting observables and their generalizations to three or more non-commuting observables. Jensen's inequality, in turn, is helpful in deriving various the "sum uncertainty relations" for two or more observables. Using these inequalities, we derive various types of generalized uncertainty relations for more than two non--commuting observables and analyze their properties and critical points. We also study the connections between the generalizations of the HR and SR uncertainty relations for two and more observables and the correlations of these observables in the state of the quantum system under study. In this analysis, we pay special attention to the consequences of the generalized SR uncertainty relation for three non--commuting observables on their correlations in a given state of a quantum system and to the connections of this relation with the appropriate correlation matrix, whose matrix elements are the quantum versions of the Pearson coefficient. We show also that the SR uncertainty relation (including the generalized ones) can be written in an equivalent way using these Pearson coefficients.

Brian P. Anderson

Júlio Glauber Ferreira dos Santos

This paper explores initially, in a intuitive form, some questions of non-commutativity in space well as the motivations for the study of non-commutative theories. Then be given an overight in some aspects of quantum theory non-commutative addressed in the plan. Following the basic rules of the non-commutative, see how to build states that saturate the uncertainty relations, i.e., get the least possible uncertainty in the observable in question, for a construction of non-commutative algebra of coordinates. In particular, following the search of states that saturate both relations Heisenberg uncertainty modied and show saturation of the Schrodinger's determinant for a given Gaussian state. Then, will be built the propagator for the free particle from the construction of noncommutative version of the plane wave, exploring the idea of average in coherent states. And also exploring the idea of the construction of wave packet in the non-commutative plane, get its evolved form using the path integral formalism, ending with the analysis of the Fourier's transform in non-commutative space. This paper analyzes the results and what can be done in a next project, with the tools presented.

Burcu Elif Akten