Energy levels, Gaussian distribution

J. Stockmann, 1999). The main achievement of this field is the establishment of universal statistics of energy levels the typical distribution of the spacing of neighbouring levels is Poisson or Gaussian ensembles for integrable or chaotic quantum systems. This statistics is well described by random-matrix theory (RMT). It was first introduced by... 

In Fig. 1 level spacing distributions for different temperatures are plotted (w = 0.01 and 9) for the energy spectrum calculated by diagonalizing of the matrix R. It is clear from this plot that the system is regular at 9 = 0. However, the increase of temperature leads to a chaotization of the system and P(S) becomes closer to the Gaussian distribution. 

As is shown in Eqns. 2-48 and 2-49, the probability density W(e) of electron energy states in the reductant or oxidant particles is represented as a normal distribution function (Gaussian distribution) centered at the most probable electron level (See Fig. 2-39.) as expressed in Eqns. 8-10 and 8-11 ... 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

Fermi function for the metal, n(E), (b) a Gaussian distribution of energy levels for acceptor states in the monolayer, Dox(E), and (c) a probability factor describing electron tunneling at a given energy, P(E) ... 

The energetics of such atomic motion can be investigated. If the probability density function is a Gaussian function, the potential energy in which the atom vibrates will be isotropic and harmonic and will have a normal Boltzmann distribution over energy levels. This potential energy will have the form ... 

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