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The phase-space formulation is a formulation of quantum mechanics that places the position and momentum variables on equal footing in phase space.The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product.
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the phase space usually consists of all possible values of the position and momentum parameters.
Phase-space representation of quantum state vectors is a formulation of quantum mechanics elaborating the phase-space formulation with a Hilbert space. It "is obtained within the framework of the relative-state formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system.
Each phase space consists of position and momentum, whose possible values are taken from a locally compact Abelian group and its dual. A quantum mechanical state can be fully represented in terms of either variables, and the transformation used to go between position and momentum spaces is, in each of the three cases, a variant of the Fourier ...
Smoothing the Wigner distribution through a filter of size larger than ħ (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.
The Glauber–Sudarshan P representation is a suggested way of writing down the phase space distribution of a quantum system in the phase space formulation of quantum mechanics. The P representation is the quasiprobability distribution in which observables are expressed in normal order.
The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector , the state vector . If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
The Husimi Q representation, introduced by Kôdi Husimi in 1940, [1] is a quasiprobability distribution commonly used in quantum mechanics [2] to represent the phase space distribution of a quantum state such as light in the phase space formulation. [3] It is used in the field of quantum optics [4] and particularly for tomographic purposes.