Wigner distributions

Lorentz, Cauchy, or Breit-Wigner distribution h(r) = j (nb ( 1 + (r-r0)2 /b2 at r0 with a full width at half-maximum b. 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

Figure 10. Excited-state population of ethylene as a function of time in femtoseconds (full line). (Results are averaged over 10 initial basis functions selected from the Wigner distribution for the ground state in the harmonic approximation.) Quenching to the ground electronic state begins approximately 50 fs after the electronic excitation, and a Gaussian fit to the AIMS data (dashed line) predicts an excited-state lifetime of 180 fs. (Figure adapted from Ref. 214.)...

This equation can be interpreted as the drift term of a collisionless Boltzmann equation for the one-particle Wigner distribution p(q,p). To see that, let us explore the physical meaning of p(q,p) in this context. First note that p(q, p ) is in principle a Lorentz scalar. Thus an invariant solution of Eq. (59) is... 

An interpolating function between the Poisson and the Wigner distribution is given by the Brody distribution (Brody, 1973) reading for the... 

Remarkably, the Wigner distribution could be observed in a number of systems by physical experiments and computer simulations evading the whole quantum world from atomic nuclei to the hydrogen atom in a magnetic field to the metal-insulator transition (Guhr, Muller-Groeling and Weidenmuller, 1998). In this contribution we address the situation in QCD and in hadrons. 

Figure 8. Histograms of the nearest-neighbor spacing distribution for the nucleon (left plots) and the delta (right plots). The data is for Goldstone-boson exchange and for one-gluon exchange compared to a pure linear confinement potential of the same strength. Curves represent the Poisson and the GOE-Wigner distributions.

For example, the vibrational initial state may be represented by a Wigner distribution (17), while the initial electronic coefficients may be determined by <4(0) = q being an arbitrary phase [200]. 

Wigner distribution function ). Show that this is not a probability density since it may become negative - e.g., when (x) = (sin x)/x /n. 

We now rewrite Eq. (2.7) by introducing the Wigner distribution for the external field as well,... 

Here the time ordering of the and 2 fields can be arbitrary we only assume that the field 3 comes after and 2 and does not overlap with them. We can then follow the calculations of pump-probe signals in Appendix E and introduce the joint Wigner distribution for the fields and 2 and for the field 3 and 4 ... 

B. Equation of Motion for the Density Operator and the Wigner Distribution... 

Thus we get the Wigner distribution by Fourier transformation with respect to y... 

The Wigner distribution function f(rpt) has many properties in common with the classical distribution function. Unfortunately, however, this distribution function is not positive definite, and, therefore, it cannot be interpreted as a usual distribution function. 

The essential information about transport properties in many-particle systems is given by the single-particle density matrix or by the singleparticle Wigner distribution. The equations of motion (1.18) and (1.23) for these important quantities are called kinetic equations. For the further consideration we write the latter equation in the momentum representation ... 

The fluctuations in neutron peak intensities arise from the Porter-Thomas distributed beta decay widths to levels in the NE nuclide. In the simplest case only a single state in the GC nuclide can be fed and only one neutron partial wave is significant. The observed levels will be a subset of levels in the NE nuclide and will be distributed in energy following a Wigner distribution. In a typical GC nuclide, however, there will be a number of accessible final states and the delayed neutron spectrum will be a superposition of transitions from several parts of the NE nuclide level structure. 

The quantum mechanical definition of a distribution function in the classical phase-space is an old theme in theoretical physics. Most frequently used is the so-called Wigner distribution function (Wigner 1932 Hillery, O Connell, Scully, and Wigner 1984). Let us consider a onedimensional system with coordinate R and corresponding classical momentum P. The Wigner distribution function is defined as... 

The Wigner distribution function for the vibrational ground state of the harmonic oscillator is the product of two Gaussians, one Gaussian in P-space centered at the equilibrium distance Re and one Gaussian in P-space localized at P = 0. 

Fig. 5.2. Contour plots of two representative Wigner distribution functions PW(R,P) for two harmonic oscillators in their ground vibrational states, Equation (5.15), in the two-dimensional phase-space (R,P). The widths in the R-and in the P-directions are inversely related.

If the parent molecule is described by normal modes with coordinates q and momenta pk j the multi-dimensional wavefunction is simply a product of uncoupled one-dimensional harmonic wavefunctions ip iQk) (Wilson, Decius, and Cross 1955 ch.2 Weissbluth 1978 ch.27) and the corresponding Wigner distribution function reads... 

The six-dimensional Wigner distribution function for the ABC molecule in its lowest state is then a product of six Gaussians,... 

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