The Wigner function

Therefore, the Wigner function is directly related to the density matrix operator, p, which characterizes the quantum system [28] in Hilbert space. 

The Wigner function is a distribution for the position q) and momentum (p) of a system. From the knowledge of the Wigner function of a system, its density matrix can be determined in a kind of quantum state tomography. 

The Wigner function defined above describes continuous systems. Until a few yeas ago, the definition of Wigner functions for discrete systems, like spin systems, was an open 

For bosons, n should run from - d - l)/2 to [d - l)/2, taking integers values between them, and from —d/2 to d — )I2, taking half odds and integers values for fermions. However, in the fermionic systems, a convention establishes that the density matrix elements of half odd should be taken as zero. 

For spin systems, the position eigenkets and eigenvalues play the role of the spin component along the z direction whereas the equivalent for momentum play the role of the quantum phases. They are connected through the discrete Fourier Transform  

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A number of procedures have been proposed to map a wave function onto a function that has the form of a phase-space distribution. Of these, the oldest and best known is the Wigner function [137,138]. (See [139] for an exposition using Louiville space.) For a review of this, and other distributions, see [140]. The quantum mechanical density matrix is a matrix representation of the density operator... 

In the quantum case this function is to be replaced by its quantum counterpart, the Wigner function [Feynman 1972 Garg et al. 1985 Dakhnovskii and Ovchinnikov 1985] expressed via the density matrix as... 

Another example of slight conceptual inaccuracy is given by the Wigner function(12) and Feynman path integral(13). Both are useful ways to look at the wave function. However, because of the prominence of classical particles in these concepts, they suggest the view that QM is a variant of statistical mechanics and that it is a theory built on top of NM. This is unfortunate, since one wants to convey the notion that NM can be recovered as an integral part of QM pertaining to for macroscopic systems. 

C. Leforestier, J. Chem. Phys., 101, 7357 (1994). Grid Method for the Wigner Functions. 

The master equation (62) can be transformed to a c-number partial differential equation. Three kinds of equations can be derived from (62) (1) an equation for the Wigner function (sym) related to symmetric (Weyl) ordering of... 

The transmission function is expressed in terms of the Wigner functions for the gates,... 

Certain semiclassical properties involving the eigenfunctions can also be calculated with periodic-orbit theory. Considering the Wigner functions corresponding to die energy eigenfunctions H = Enn [28],... 

P. R. Holland, A. Kyprianidis, Z. Marie, and J. P. Vigier, Relativistic generalization of the Wigner function and its interpretation in the causal stochastic formulation of quantum mechanics, Phys. Rev. A (Special Issue General Physics) 33(6), 4380—4383 (1986). 

Thus the results for all three coupling schemes are given by the same kind of expression involving the Wigner function, differing only in the angle. 

Dahl, J.P. (1983). Dynamical equations for the Wigner functions, in Energy Storage and Redistributions in Molecules, ed. J. Hinze (Plenum Press, New York). 

Quantum and Semiclassical Approaches A. The Wigner Function and Weyl s Rule Quantum Scars in Phase-Space Quantizing the ARRKM Theory... 

The phase space structure of classical molecular dynamics is extensively used in developing classical reaction rate theory. If the quanmm reaction dynamics can also be viewed from a phase-space perspective, then a quantum reaction rate theory can use a significant amount of the classical language and the quantum-classical correspondence in reaction rate theory can be closely examined. This is indeed possible by use of, for example, the Wigner function approach. For simplicity let us consider a Hamiltonian system with only one DOF. Generalization to many-dimensional systems is straightforward. The Wigner function associated with a density operator /)( / is defined by... 

CLASSICAL, SEMICLASSICAL, AND QUANTUM MECHANICAL RATE THEORY 105 The Wigner function thus defined has the following property [68] ... 

Within the Wigner function framework the operator exp iaq + ibp) is associated with the phase-space function exp iaq + ibp). This particular association between a phase-space function and a function of noncommuting operators is an example of WeyTs rule [69]. Indeed, by generalizing Eq. (1.317) one obtains the Weyl transform Aw(q,p) of an arbitrary operator... 

The Wigner function has the valuable property that the time evolution equation for the quantum dynamics in the Wigner representation resembles that for the classical Liouville dynamics. Specifically, the Schrodinger equation can be transformed to [70]... 

The fact that classical unstable periodic trajectories can manifest themselves in the Wigner function implies that nonstatistical behavior in the quanmm dynamics can be intimately related to the phase-space structure of the classical molecular dynamics. Consider, for example, the bottlenecks to intramolecular energy flow. Since the intramolecular bottlenecks are caused by remnants of the most robust tori, they are presumably related to the least unstable periodic trajectories. Hence quantum scars, being most significant in the case of the least unstable periodic trajectories, are expected to be more or less connected with intramolecular bottlenecks. Indeed, this observation motivated a recent proposal [75] to semiclassically locate quantum intramolecular bottlenecks. Specifically, the most robust intramolecular bottlenecks are associated with the least unstable periodic trajectories for which Eq. (332) holds, that is,... 

To visualize the time-evolving Wigner function associated with the reaction coordinate, we choose r = 5.70 a.u., which is close to the minimum of the I2 potential energy curve. The Wigner functions at four different times are displayed in Fig. 46. From Fig. 46 it is seen that as time evolves, more and more phase space flux flows out of the original support of the Wigner function. Also... 

Finally, for a reactant system including several vibrational modes, the Wigner function is simply a product of one dimensional functions ... 

Wc have taken the initial coordinates and momenta for the swarm of trajectories from the Wigner distribution. It can be easily proven that for a separable wavefunction the Wigner function can be written in a product form. As a basis for photodissociation calculations we have used a wavefunction in the form 3, therefore, the corresponding Wigner distribution is given by an expression... 

Suppose that the interaction of the molecule with the dissipative medium is realized through the rotational-only degrees of freedom of the molecule and does not affect the vibronic ones. Then, passing to the system of coordinates linked to the molecule by means of the Wigner functions Djmm and using the classical description for the rotational diffusion, we have... 

Writing down the time evolution equation for the Wigner function, we find from the Master Equation (26)... 

For simulations, we have chosen coherent amplitudes (3 = 2 and f3 = 4, see Fig. 2, in order to compare the decoherence rate, and an initial ancilla state given by z = 0.5, thus avoiding eigenstates of Pauli matrices. Figs. 2 to 4 show the Wigner function... 

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