Wigner functions

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... 

A final study that must be mentioned is a study by Haitmann et al. [249] on the ultrafast spechoscopy of the Na3p2 cluster. They derived an expression for the calculation of a pump-probe signal using a Wigner-type density mahix approach, which requires a time-dependent ensemble to be calculated after the initial excitation. This ensemble was obtained using fewest switches surface hopping, with trajectories inibally sampled from the thermalized vibronic Wigner function vertically excited onto the upper surface. 

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. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

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 first term in Eq. (4.3) is reminiscent of Eq. (3.2) for the spontaneous emission spectrum. It represents a doorway wavepacket created by the pump in the excited state, which is then detected by its overlap with a window. The only difference is that the role of the gate in determining the window in SLE is now played by the probe Wigner function W2. In addition, the pump-probe signal contains a second term that does not show up in fluorescence. This term represents a wavepacket created in the ground state (a hole ) that evolves in time as well and is finally determined by a different window Wg [24]. In the snapshot limit, defined in the preceding section, we have... 

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). 

Expansion (7.58) becomes obvious if we consider the lattice symmetry. Two nearest neighbors belonging to the same sublattice are situated on the diagonals of the opposite vertices and their contributions to the crystalline field should be equal, so that the odd harmonics vanish. Coefficients /34 and j86 can be calculated using the AAP method. Yamamoto et al. [1977] found /34 = 5.43, p6 = 7.18 these numbers were later revised by Smith [1990] so as to give the best fit to the libration spectrum (/34 = 6.50, /36 = -5.33). The intermolecular potential is expanded in a series in rotational tetrahedral functions uv, which can be expressed in terms of Wigner functions at 7 = 3 ... 

The coefficient <57 and the coefficients at rotational Wigner functions entering into (j>a) are found by solving self-consistent equations (7.60). According to [Smith, 1990], 57 is equal to 19.65 for CH4 and 48 for CD4. If one neglects the crystalline field potential Vc [Smith, 1985 Huller and Raich, 1979], then the Oh molecules rotate freely, as follows from (7.62). 

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). 

---