Spacing levels

The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

The 70 years since these first observations have witnessed dramatic developments in Raman spectroscopy, particularly with the advent of lasers. By now, a large variety of Raman spectroscopies have appeared, each with its own acronym. They all share the conunon trait of using high energy ( optical ) light to probe small energy level spacings in matter. 

When the potential V Q) is symmetric or its asymmetry is smaller than the level spacing (Oq, then at low temperature (T cuo) only the lowest energy doublet is occupied, and the total energy spectrum can be truncated to that of a TLS. If V Q) is coupled to the vibrations whose frequencies are less than coq and co, it can be described by the spin-boson Hamiltonian... 

Figure 15. The discrete energy levels in Au55(PPh3)i2Cl6 at 7 K. The level spacing is 135mV. (Reprinted from Ref [24], 2003, with permission from American Chemical Society.)...

Discrete energy levels are to be observed for position (a) as well as for position (b) at exactly the same values, in case (b) somewhat better expressed than in (a). The level spacing is 135 mV. This spectrum clearly identifies the Au55 cluster as a quantum dot in the classical sense, having discrete electronic energy levels, though broader than in an atom, but nevertheless existent. The description of such quantum dots as artificial, big atoms seems indeed to be justified. 

Figure 10. Level spacing distributions P(s/ s)) for the cone states of the first-excited electronic doublet state of Li3 with consideration of GP effects [12] (a) A symmetry (b) A2 symmetry (c) E symmetry (d) full spectrum. Also shown by the solid lines are the corresponding fits to a Poisson distribution.

Vibrational spectroscopy Calculation of diatomic energy level spacings, isotope shifts... 

In order to understand these observations it is necessary to resort to quantum mechanics, based on Planck s postulate that energy is quantized in units of E = hv and the Bohr frequency condition that requires an exact match between level spacings and the frequency of emitted radiation, hv = Eupper — Ei0wer. The mathematical models are comparatively simple and in all cases appropriate energy levels can be obtained from one-dimensional wave equations. 

Another development in the quantum chaos where finite-temperature effects are important is the Quantum field theory. As it is shown by recent studies on the Quantum Chromodynamics (QCD) Dirac operator level statistics (Bittner et.al., 1999), nearest level spacing distribution of this operator is governed by random matrix theory both in confinement and deconfinement phases. In the presence of in-medium effects... 

Figure 1. The nearest neighbor level spacing distribution for various parameter of temperature a) zero temperature case b) (3 = 0.1 c) (3 = 0.01 ...

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. 

Abstract. Quantum chaos at finite-temperature is studied using a simple paradigm, two-dimensional coupled nonlinear oscillator. As an approach for the treatment of the finite-temperature a real-time finite-temperature field theory, thermofield dynamics, is used. It is found that increasing the temperature leads to a smooth transition from Poissonian to Gaussian distribution in nearest neighbor level spacing distribution. 

One of the main characteristics of the statistical properies of the spectra is the level spacing distribution (Eckhardt, 1988 Gutzwiller, 1990) function. In this work we calculate the nearest-neighbor levelspacing distribution (Eckhardt, 1988 Gutzwiller, 1990). 

The nearest neighbor level spacings are defined as Si = -E)+i — Ei, where Ei are the energies of the unfolded levels, which are obtained by the following way The spectrum Ei is separated into smoothed average part and fluctuating parts. Then the number of levels below E is counted and the following staircase function is defined ... 

Then the nearest level spacing distribution function P S) is defined as the probability of S lying within the infinitesimal interval [S, S+dS]. 

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. 

Figure 1. The level spacing distributions for different temperatures a) 6 (zero-temperature case) b) 6 = 0.01 c) 0 = 0.2 at the fixed lo = 0.01.

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