Quantum distributions

The quantum analogs of the phase space distribution function and the Lionville equation discussed in Section 1.2.2 are the density operator and the quantum Lionville equation discussed in Chapter 10. Here we mention for future reference the particularly simple results obtained for equilibrium systems of identical noninteracting particles. If the particles are distinguishable, for example, atoms attached to their lattice sites, then the canonical partitions function is, for a system of N particles 

Using Eq. (1.161), both Eqs (1.171) and (1.172) lead to the same expression forthe average system energy 

At low temperature the situation is complicated by the fact that the difference between distinguishable and indistinguishable particles enters only when they occupy different states. This leads to different statistics between fermions and bosons and to the generalization of (1.174) to 

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For many applications, it may be reasonable to assume that the system behaves classically, that is, the trajectories are real particle trajectories. It is then not necessary to use a quantum distribution, and the appropriate ensemble of classical thermodynamics can be taken. A typical approach is to use a rnicrocanonical ensemble to distribute energy into the internal modes of the system. The normal-mode sampling algorithm [142-144], for example, assigns a desired energy to each normal mode, as a harmonic amplitude... 

This relationship defines the following three equilibrium quantum distributions... 

The atoms in a molecule are never stationary, even close to the absolute zero temperature. However the physical scale of the vibrational movement of atoms in molecules is rather small - of the order of 10 to 10 ° cm. The movement of the atoms in a molecule is confined within this narrow range by a potential energy well, formed between the binding potential of the bonding electrons, and the repulsive (mainly electrostatic) force between the atomic nuclei. Whenever atomic scale particles are confined within a potential well, one can expect a quantum distribution of energy levels. 

The analysis of the classical dynamics shows a transition to chaotic motion leading to diffusion and ionization [6]. In the quantum case, interference effects lead to localization and the quantum distribution reaches a steady state that is exponentially localized (in the number of photons) around the initially excited state. As a consequence, ionization will take place only when the localization length is large enough to exceed the number of photons necessary to reach the continuum. 

Figure 2 shows in the full curves the relative fluorescence spectrum of PST with the maximum at 32,000 cm"1 and of 10"4 mole % TPB in PST with the maximum at 23,000 cm"1. The spectra were drawn with maxima of the same height. The dashed curves show the spectral transmittance of the same samples. As seen from the overlapping of transmittance and the fluorescence quantum distribution, part of self-absorption is large, whereas it is much smaller in the case of TPB fluorescence. 

The classical calculations reproduce — on the average — the quantum distributions quite well. At very low energies the classical distributions... 

O Cormell, R.F. and E.P. Wigner, Manifestations of Bose and Fermi statistics on the quantum distribution function for systems of spin-0 and spin-1/2 particles. Physical Review A, 1984. 30(5) p. 2613-2618. 

The quantum distribution of electrons in metals has a profound effect on many of their properties. As an example consider their contribution to a metal heat capacity. 

Quantum distributions, denoted p (p,q), do not have a probabilistic interpretation. Rather, the conditions closest to those in Eq. (3.31) are... 

Abstract We review the basic theoretical formulation for pulsed X-ray scattering on nonstationary molecular states. Relevant time scales are discussed for coherent as well as incoherent X-ray pulses. The general formalism is applied to a nonstationary diatomic molecule in order to highlight the relation between the signal and the time-dependent quantum distribution of intemuclear positions. Finally, a few experimental results are briefly discussed. 

After 100 ps, the anisotropy introduced by the pump laser has disappeared due to the interaction with the solvent. Thus, with the isotropy of the liquid system before and after laser excitation, the contribution to the signal from the solute can be described by an equation similar to (44), with the quantum distribution of intemuclear positions replaced by paverage(R tp), i.e., the time-dependent I-I atom-atom distribution function. 

The (quantum) distributions Fermi-Dirac and Bose-Einstein, with application to the classification of fundamental particles and forces. 

TABLE 1.4 The Illustration of a Mode (from the possible ones) for the Quantum Distribution for Fermionic Type Particles with Half-Integer Spin on an Energetic Level with Sub-Levels g. (Putz, 2010)... 

The first level of unification of distributions consists in reducing the FD and BE quantum distribution to the Boltzmaim one s, for system achieving large mass for particles and in any case much largerthanthe one of the electron/fermions and of the photon/bosons in moving at high temperatures (classical mode). Analytically, the first condition is rendered into... 

The second level of unification of quantum distributions, is more subtle and relates to (i) the quality of fermions (half-integer spin) to characterize the substance elementary particles of matter (ii) the bosons (integer spin) as particles associated to the fundamental fields (to the forces implicitly) of matter that intermediate the interactions between the elementary particles. For clarity, we present in Table 1.6, face-to-face, the elementary particles for substance and the characteristic particle-carriers to the fundamental forces of Nature. 

Molecular treats discrete entities quantum distribution functions collision integrals... 

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