Particle, operational definition

This confirms our interpretation of the operators 6,6 and d,d as creation and annihilation operators for particles of definite momentum and energy. Similar consideration can be made for the angular momentum operator. The total electric charge operator is defined as... 

If we expand the in and out operators in terms of operators creating particles of definite momentum, i.e.,... 

However, these results are probably the result of the method of detecting particles and thereby provide an operational definition of "particle". All of these studies use a condensation nuclei counter to determine the presence of "particles". These devices have been found to have sharply decreasing efficiency for detecting particles less than 0.01 ym (Leong, et al., 1983, for example). 

Fiber has been given a legal, if not operational definition, at least as it applies to asbestos. Unfortunately tlie definition bears little relationship to the present use of the term. It was essential to set standards to reduce occupational exposure but the detection, identification, and suppression of asbestos materials opened several areas of problems that remain unsolved. For example, six minerals are included in the definition of asbestos. Are they all equally hazardous If not, why not There are many other inorganic particles with diameters of less than 3 micrometers and a diameter-to-length ratio of 1 3. Should we be concerned that they too might be hazardous to our health ... 

The unit tensor operators are irreducible-tensor operators with reduced matrix elements of unity. They are a valid choice to use as a basis in order to express any arbitrary tensor operator as a linear combination, since they are linearly independent. Attention is restricted to these for the sake of simplicity. Hence the definition of unit tensor single-particle operator U (aK, a L, r) is... 

A system of solid particles dispersed in a liquid. Suspensions were previously referred to as suspensoids , meaning suspension colloids. Aside from the obvious definition of a colloidal suspension, a number of operational definitions are common in industry, such as any dispersed matter that can be removed by a 0.45 pm nominal pore-size filter. 

An operational definition is considerably more practical. Operationally determined species are defined by the methods used to separate them from other forms of the same element that may be present. The physical or chemical procedure that isolates the particular set of metal species is used to define the set. Metals extracted from soil with an acetate buffer is an operational definition of a certain class. Lead present in airborne particles of less than 10 pm is another. In water analyses, simply filtering the sample before acidification can speciate the analytes into dissolved and insoluble fractions. These procedures are sometimes referred to as fractionation, which is probably a more properly descriptive term than speciation, as speciation might imply that a particular chemical species or compound is being determined. When such operational speciation is done, careful documentation of the protocol is required, since small changes in procedure can lead to substantial changes in the results. Standardized methods are recommended, as results cannot be compared from one laboratory to another unless a standard protocol is followed [124], Improvements in methodology must be documented and compared with the currently used standard methods to produce useful, readily interpretable information. 

Shape is defined as the pattern of relationships among all of the points of the surface or profile of the particle. The development of this operational definition has in turn led to the development of morphological analysis. 

Trace metals in seawater can exist in a variety of physical and chemical forms. The simplest physical distinction is particulate versus dissolved forms. This is somewhat of an operational definition with 0.4 p.m or 0.2 p.m pore size filters generally providing this separation. Particulate forms include those metals adsorbed onto particle surfaces, incorporated within particles of biogenic origin and incorporated in the matrix of... 

This new definition of normal ordering changes our analysis of the Wick s theorem contractions only slightly. Whereas before, the only nonzero pairwise contraction required the annihilation operator to be to the left of the creation operator (cf. Eq. [84]), now the only nonzero contractions place the q -particle operator to the left of the -particle creation operator. There are only two ways this can occur, namely. 

We assume that Rk can be expressed in terms of the operators 0 which are strings of quasi-particle operators Yp, where, by definition, Y° = Xa... 

Equation (4.14) provides the equivalence between the dipole length and dipole velocity transition moments for a system of n identical particles of mass m with state-independent effective operator definition A. To see that this equivalence does not produce a sum rule, consider first the usual derivation of the Thomas-Reiche-Kuhn sum rule for the true operators. Left- and right-multiplying equation (4.12) by ( l and ), respectively, the z component yields... 

Note that expressions like two-particle operator and single-particle Green s function refer to the number of particles involved in the definition of the operator (two in the case of an interaction, one for a potential energy, etc.), not to the total number of particles present in the system. 

If applied to the reference state normal order enables us immediately to recognize those terms which survive in the computation of the vacuum amplitudes. The same applies for any model function and, hence, for real multidimensional model spaces, if a proper normal-order sequence is defined for all the particle-hole creation and annihilation operators from the four classes of orbitals (i)-(iv) in Subsection 3.4. In addition to the specification of a proper set of indices for the physical operators, such as the effective Hamiltonian or any other one- or two-particle operator, however, the definition and classification of the model-space functions now plays a crucial role. In order to deal properly with the model-spaces of open-shell systems, an unique set of indices is required, in particular, for identifying the operator strings of the model-space functions (a)< and d )p, respectively. Apart from the particle and hole states (with regard to the many-electron vacuum), we therefore need a clear and simple distinction between different classes of creation and annihilation operators. For this reason, it is convenient for the derivation of open-shell expansions to specify a (so-called) extended normal-order sequence. Six different types of orbitals have to be distinguished hereby in order to reflect not only the classification of the core, core-valence,... orbitals, following our discussion in Subsection 3.4, but also the range of summation which is associated with these orbitals. While some of the indices refer a class of orbitals as a whole, others are just used to indicate a particular core-valence or valence orbital, respectively. 

Metals associated with various binding sites on sediments have been assessed using extraction procedures applied as single digests or as a set of sequential steps. Selective dissolution of trace metals from the particle surfaces is followed by determination using atomic absorption spectrometry (AAS), ICP-MS, or total reflection X-ray fluorescence. The use of sequential extraction schemes for the operational definition of metal species in sediments has proved contentious. They have been criticized on the basis that the reactions are not sufficiently phase selective and labile phases could be transformed during sample preparation, causing a marked reduction... 

Chemists are more used to the operational definition of symmetry, which crystaUo-graphers have been using long before the advent of quantum chemistry. Their ball-and-stick models of molecules naturally exhibit the symmetry properties of macroscopic objects they pass into congruent forms upon application of bodily rotations about proper and improper axes of symmetry. Needless to say, the practitioner of quantum chemistry and molecular modeling is not concerned with balls and sticks, but with subatomic particles, nuclei, and electrons. It is hard to see how bodily rotations, which leave all interparticle distances unaltered, could affect in any way the study of molecular phenomena that only depend on these internal distances. Hence, the purpose of the book will be to come to terms with the subtle metaphors that relate our macroscopic intuitive ideas about symmetry to the molecular world. In the end the reader should have acquired the skills to make use of the mathematical tools of group theory for whatever chemical problems he/she will be confronted with in the course of his or her own research. 

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