Wigner statistics

To conclude this section we point out that the Wigner statistics is valid only if the system has integer spin and is invariant under an anti-unitary symmetry such as time reversal symmetry. If all the antiunitary symmetries are broken, the nearest neighbour statistics is expected to be... 

The emergence of Wigner statistics can be approached from the point of view of ab initio multiconfigurational Dirac-Fock theory. Although a full calculation for the 5p spectrum of Ba presents a formidable challenge and has not yet proved possible, fairly extensive calculations have proved fea-... 

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. 

Hillery, M., R.O. Connell, M.O. Scully and E.P. Wigner. Phys. Rep., 106 121, 1984. Zwanzig, R. Nonequilibrium Statistical Mechanics. Oxford University Press, Oxford, 2001. 

Taking the experimentally measured mass spectrum of hadrons up to 2.5 GeV from the Particle Data Group, Pascalutsa (2003) could show that the hadron level-spacing distribution is remarkably well described by the Wigner surmise for / = 1 (see Fig. 6). This indicates that the fluctuation properties of the hadron spectrum fall into the GOE universality class, and hence hadrons exhibit the quantum chaos phenomenon. One then should be able to describe the statistical properties of hadron spectra using RMT with random Hamiltonians from GOE that are characterized by good time-reversal and rotational symmetry. 

Transition state theory, a quasi-thermodynamic/statistical mechanical approach to the theory of reaction rates was developed in the early 1930s by a number of workers including H. Eyring, E. R Wigner, and J. C. Polanyi and was very quickly applied to the consideration of isotope effects on rates of simple molecular reactions. 

Eb are the energy barriers for forward and reverse reactions, A Hr is the heat of the reaction to be discussed later, and the horizontal scale is called the reaction coordinate, an iU-defmed distance that molecules must travel in converting between reactants and products. Polanyi and Wigner lirst showed from statistical mechanics that the rates should be described by expressions of the form as given in the boxed equation by a Boltzmann factor, exp( —E / R T), which is the probabihty of crossing a potential energy barrier between reactant and product molecules. In fact, it is very rare ever to fmd reaction-rate coefficients that are not described with fair accuracy by expressions of this form. 

The spin statistical factor is found from Wigner s spin rules in this case, a — n/3m. Recent work has been discussed by Wilkinson and Tsiamis... 

K. Imre, E. Ozizmir, M. Rosenbaum, and P. F. Zweifel. Wigner method in quantum statistical mechanics. Journal of Mathematical Physics, 8(5) 1097-1108, 1967. 

Imre, K., Ozimir, E., Rosenbaum, M., Zweifel, P.F. Wigner method in quantum statistical mechanics. J. Math. Phys. 8 1097 (1967). 

As a consequence of the Wigner-Eckart theorem, relations between statistical tensors which can be derived from purely vector coupling procedures will be supplemented for transitions by introducing the corresponding reduced matrix elements Dy or Cy for the process of photoionization or Auger decay, respectively (see equs. (8.102) and (8.103b)). 

The random matrix was first introduced by E. P. Wigner as a model to mimic unknown interactions in nuclei, and it has been studied to describe statistical natures of spectral fluctuations in quantum chaos systems [17]. Here, we introduce a random matrix system driven by a time-dependent external field E(t), which is considered as a model of highly excited atoms or molecules under an electromagnetic field. We write the Hamiltonian... 

One of us has personally applied the theory of Wigner functions to obtain statistical weights for individual trajectories in the study of the photochemical dissociation of the water molecule in its second singlet excited state ... 

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