ua nt - p h / 04 11 02 8 v 1 4 N ov 2 00 4 Bounds on integrals of the Wigner function : the hyperbolic case
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چکیده
I. Abstract Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly-solvable eigenvalue problem and that the bounds differ from those on classical Liouville distributions. In particular, we show that the total " quasiprobability " on such a region can be greater than 1 or less than zero. Since its introduction 1 , the Wigner function has been the subject of extensive study in the fields of quantum physics, quantum chemistry and signal analysis (see and references therein). Since Wigner functions represent quantum states on phase space, they play a key role in the phase space formulation of quantum mechanics. They are also designed to closely resemble the joint densities of position and momentum, known as Liouville 1
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تاریخ انتشار 2008