Quantization, definition

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

A quantization of the z, variables resulted in the definition of the following categorical performance variables, y, ... 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

All spectra are due to the absorbance of electromagnetic radiation energy by a sample. Except for thermal (kinetic) energy, all other energy states of matter are quantized. Quantized transitions imply precise energy levels that would give rise to line spectra with virtually no line-width. Most spectral peaks have a definite width that can be explained in several ways. First, the spectral line-width can be related to the... 

The rotational levels are sketched for two lowest vibrational levels, v = 0 and v = 1. Since vibrational and rotational levels are quantized just like electrons, therefore, for a given molecular species these energy changes will have certain definite values. 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

The reduced matrices and V represent a partitioning of the Hamiltonian into one- and two-electron parts. Rearranging the second-quantized operators and using the definition of the 2- and 3-RDMs,... 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

The term quantization or quantum configuration means here the kind of subdivision of the electronic system into definite groups (quanticules). For instance, Na+Cl- contains two mononuclear quanticules 1 2 and 1 2 3 , while H+ejH+ contains one binuclear quanticule e. See Fajans, K. Chimia 13, 349 (1959). Lithoprinted English translation at Ulrich s Book Store, Ann Arbor, Michigan. 

Let us start this section by recalling the general definition of a p-RDM in second quantization ... 

The detection of sharp plasmon absorption signifies the onset of metallic character. This phenomenon occurs in the presence of a conduction band intersected by the Fermi level, which enables electron-hole pairs of all energies, no matter how small, to be excited. A metal, of course, conducts current electrically and its resistivity has a positive temperature coefficient. On the basis of these definitions, aqueous 5-10 nm colloidal silver particles, in the millimolar concentration range, can be considered to be metallic. Smaller particles in the 100-A > D > 20-A size domain, which exhibit absorption spectra blue-shifted from the plasmon band (Fig. 80), have been suggested to be quasi-metallic [513] these particles are size-quantized [8-11]. Still smaller particles, having distinct absorption bands in the ultraviolet region, are non-metallic silver clusters. 

Most formulations of MCSCF theory are based on the second quantization formalism. We therefore review briefly in this section the basic definitions of the annihilation and creation operators, and the expansion of quantum mechanical operators in products of them. 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

Note, that this phase factor differs by (—l)2 from the phase factor that defines the standard phase system (Eq. (4) in the Introduction). This difference manifests itself only at half-integer values of momentum j. This change in the definition of the standard phase system is caused by the presence of commonly accepted phase relations in the second-quantization method [12, 96]. 

Atomic spectral moments can be expressed in terms of the averages of the products of the relevant operators. Let Oi,C>2, --,Ok be the operators of interactions in definite shells or the operators of electronic transitions between definite shells in the second quantization form. The average of... 

In this tribute and memorial to Per-Olov Lowdin we discuss and review the extension of Quantum Mechanics to so-called open dissipative systems via complex deformation techniques of both Hamiltonian and Liouvillian dynamics. The review also covers briefly the emergence of time scales, the definition of the quasibosonic pair entropy as well as the precise quantization relation between the temperature and the phenomenological relaxation time. The issue of microscopic selforganization is approached through the formation of certain units identified as classical Jordan blocks appearing naturally in the generalised dynamical picture. 

These are quantized vibrations of conduction electrons in a metal or semiconductor 4°). The quantized energy levels of the collective longitudinal vibrations of the electron gas are quasi-particles in the sense of the definition given above and are called plasmons. The frequency of this longitudinal vibration, the plasma frequency u>P, is given by 40>... 

In the present and in the following section we discuss the application of the group-theoretical formalism to the formulation of quantum-classical mechanics. Our purpose is to determine evolution equations for two coupled subsystems, with two different degrees of quantization. We have shown in the previous sections that the classical behaviour of a system is formally obtained as a limiting case of the quantum behaviour, when the Planck constant h tends to zero. In this section we will associate two different values of the Planck constant, say hi and /12, to the two subsystems and introduce suitable Lie brackets to determine the evolution of the two subsystems [15]. The consistency, e.g., with respect to Jacobi identity, is guaranteed by the very definition of the... 

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