Wigner distribution spacing

Here, we give here a brief outline of the methods as introduced in Refs. 43, 44, and 47. Suppose that the initial state of the system is tpoili, ,<]n) From ipo, the Wigner phase-space distribution D(qi,..., qn pi,. ..,Pn] is computed. This distribution is used to sample initial positions and momenta, . .., for a classical trajectory simulation of the process of... 

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.

D. J. Tannor To understand the role of dissipation in quantum mechanics, it is useful to consider the density operator in the Wigner phase-space representation. Energy relaxation in a harmonic oscillator looks as shown in Fig. 1, whereas phase relaxation looks as shown in Fig. 2 that is, in pure dephasing the density spreads out over the energy shell (i.e., spreads in angle) while not changing its radial distribution... 

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.

The mean spacing is normalized to 1 by choosing A = 7t/4. This choice yields the final result, the Wigner distribution Pw x), in the form... 

Fig. 4.4. Wigner distribution Pw x) (dotted line) and the limiting distribution Pi x) for the normalized eigenvalue spacings of M x M matrices with M —> oo (full line). The function Pl x) was plotted using the data listed in Table 4.3 of Haake (1991).

Figure 1. Distribution of spacings of nearest-neighbor vibrational energy levels of the vibrational states of Ar3 with A, symmetry, that of the ground vibrational state. The histogram shows the computed level spacings. The solid curve is that of a fitted Wigner distribution, and the dotted curve is that of a Poisson distribution. [Reprinted with permission from D. M. Leitner, R. S. Berry, and R. M. Whitnell,, /. Chem. Phys. 91, 3470 (1989). Copyright 1989, American Institute of Physics.]...

We stress that level spacings alone do not suffice to make statements about quantum chaos integrable systems are known which give Wigner distributions [561, 562]. Simple systems are also known whose classical dynamics is chaotic, but whose nearest level spacings differ markedly from the Wigner distribution [563]. For atoms, such systems can have the property that one series is dominant, i.e. has a much higher density of... 

Figure 9 Wigner distribution function of the n = 10 eigenfunction of the harmonic oscillator. The picture shows the extent of the wave function in phase space which has nearly optimal sampling due to the balance between the representation of the kinetic and potential energy.

The Wigner distribution (92) has been put on a much firmer theoretical basis by the work of Dyson 46). He derives a distribution that can be approximated by (92) to a high degree of accuracy. It has also been observed experimentally that the Wigner distribution gives a good fit to the resonance spacing. 

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