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Phase Space Picture and Canonical Transformations

S. Rajasekar, R. Velusamy

This chapter presents some of the features of the phase space picture. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture. Using the Wigner distribution function it is possible to perform a canonical transformation in quantum mechanics. The phase space picture is a useful candidate to illustrate the transition from classical to quantum mechanics. The Poisson brackets and canonical transformations are part of the effective formulation of classical mechanics. In the infinite height potential video sequences of the evolution of Wigner functions for a wave function of a wave packet and its mirror wave packet is constructed. The chapter also presents the application of the canonical transformations and phase space distribution function in the study of wave packet spread and coherent and squeezed states.This chapter presents some of the features of the phase space picture. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture. Using the Wigner distribution function it is possible to perform a canonical transformation in quantum mechanics. The phase space picture is a useful candidate to illustrate the transition from classical to quantum mechanics. The Poisson brackets and canonical transformations are part of the effective formulation of classical mechanics. In the infinite height potential video sequences of the evolution of Wigner functions for a wave function of a wave packet and its mirror wave packet is constructed. The chapter also presents the application of the canonical transformations and phase space distribution function in the study of wave packet spread and coherent and squeezed states.

CRC Press eBooksOct 5, 20220 citations

Properties of Wigner distribution functions applied to quantum mechanics

A V Gorbachev

Вестник Российского университета дружбы народов. Серия: Математика, информатика, физикаJan 1, 20120 citations

Wigner Distribution Functions in Relativistic Quantum Mechanics

J. H. Irving

Jan 1, 19650 citations

Wigner phase-space distribution as a wave function

Denys I. Bondar, Renán Cabrera, Dmitry V. Zhdanov +1 more

We demonstrate that the Wigner function of a pure quantum state is a wave function in a specially tuned Dirac bra-ket formalism and argue that the Wigner function is in fact a probability amplitude for the quantum particle to be at a certain point of the classical phase space. Additionally, we establish that in the classical limit, the Wigner function transforms into a classical Koopman--von Neumann wave function rather than into a classical probability distribution. Since probability amplitude need not be positive, our findings provide an alternative outlook on the Wigner function's negativity.We demonstrate that the Wigner function of a pure quantum state is a wave function in a specially tuned Dirac bra-ket formalism and argue that the Wigner function is in fact a probability amplitude for the quantum particle to be at a certain point of the classical phase space. Additionally, we establish that in the classical limit, the Wigner function transforms into a classical Koopman--von Neumann wave function rather than into a classical probability distribution. Since probability amplitude need not be positive, our findings provide an alternative outlook on the Wigner function's negativity.

arXiv:1202.3628Physical Review ANov 11, 201350 citations

Phase-space formulation of filtering: insight into the wave-particle duality

Daniela Dragoman

A phase-space formulation of the filtering process of a quantum state is developed that models both quantum interference and delayed-choice experiments without reference to the controversial wave-particle duality concept and without any explicit model of the interaction between the particles and the measuring apparatus. Quantum particles are considered as localized and indivisible concentrations of energy and/or mass, their probability amplitude in phase space being described by the Wigner distribution function. The "wave" or "particle" nature emerges in experiments in which the interference term of the Wigner distribution function is present or absent, respectively. The Wigner distribution function is modified by filtering devices that act on the quantum wave function throughout the setup, from its generation to its final detection.A phase-space formulation of the filtering process of a quantum state is developed that models both quantum interference and delayed-choice experiments without reference to the controversial wave-particle duality concept and without any explicit model of the interaction between the particles and the measuring apparatus. Quantum particles are considered as localized and indivisible concentrations of energy and/or mass, their probability amplitude in phase space being described by the Wigner distribution function. The "wave" or "particle" nature emerges in experiments in which the interference term of the Wigner distribution function is present or absent, respectively. The Wigner distribution function is modified by filtering devices that act on the quantum wave function throughout the setup, from its generation to its final detection.

arXiv:quant-ph/0402061Journal of the Optical Society of America BMar 1, 20053 citations

Algebraic aspects of the wigner distribution in quantum mechanics

Narasimhaiengar Mukunda

PramanaJul 1, 197824 citations

On constructing the wave function of a quantum particle from its Wigner phase-space distribution function

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