In this letter, firstly, the Schr $\\ddot{o}$ dinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space-space and space-momentum as well as momentum-momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.In this letter, firstly, the Schr$\\ddot{o}$dinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space-space and space-momentum as well as momentum-momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.

Jan 1, 20059 citations

Quantum mechanics on phase space

Nicolaas P. Landsman

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsJan 1, 19996 citations

Contracted Schrödinger equation in quantum phase‐space

C. Frishberg, Leon W. Cohen

The phase space formulation of quantum mechanics is equivalent to standard quantum mechanics where averages are calculated by way of phase space integration as in the case of classical statistical mechanics. We derive the quantum hierarchy equations, often called the contracted Schrödinger equation, in the phase space representation of quantum mechanics which involves quasi-distributions of position and momentum. We use the Wigner distribution for the phase space function and the Moyal phase space eigenvalue formulation to derive the hierarchy. We show that the hierarchy equations in the position, momentum, and position-momentum representations are very similar in structure. © 2017 Wiley Periodicals, Inc.The phase space formulation of quantum mechanics is equivalent to standard quantum mechanics where averages are calculated by way of phase space integration as in the case of classical statistical mechanics. We derive the quantum hierarchy equations, often called the contracted Schrödinger equation, in the phase space representation of quantum mechanics which involves quasi-distributions of position and momentum. We use the Wigner distribution for the phase space function and the Moyal phase space eigenvalue formulation to derive the hierarchy. We show that the hierarchy equations in the position, momentum, and position-momentum representations are very similar in structure. © 2017 Wiley Periodicals, Inc.

Journal of Computational ChemistryJul 27, 20171 citation

Quantum Mechanics on Phase Space

Manuel Gadella

Springer eBooksJan 1, 19933 citations

Systems of imprimitivity and representations of quantum mechanics on fuzzy phase spaces

S. Twareque Ali, Eduard Prugovečki

The problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so-called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.The problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so-called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.

Journal of Mathematical PhysicsFeb 1, 1977154 citations

In this letter, firstly, the Schr

\\ddot{o}

dinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space-space and space-momentum as well as momentum-momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.