Abstracts mathematical symbols defined

There are many uses for operations on sets in abstract descriptions. Because the traditional mathematical symbols are not available on most keyboards, OCL defines ASCII equivalents. It is easier to use + for union, for intersection, and - for set difference (see Section 2.4.5, Collections), and in Catalysis these features can be extended by the modeler. 

Define all symbols for mathematical constants, variables, and unknown quantities the first time you use them in the text. If you use them in the abstract, define them there and then again at their first appearance in text. Do not define standard mathematical constants such as n, i, and e. 

Even if these formulas in the beginning have a more symbolic meaning, they may be given a strict mathematical interpretation by properly defining the domains of the operators and superoperators involved. Usually the wave function space is considered as a realization of the abstract Hilbert space based on a binary product ( 1 2) of the L2-type ... 

Besides the apparent similarities. Table 8.1 illustrates also the obvious formal differences between bras and kets and their second quantized counterparts. Namely, the corresponding symbols are mathematically very different. The bra and ket vectors are elements of a linear vector space over which quantum-mechanical operators are defined, while the creation and annihilation operators are defined over the abstract space of particle number represented wave functions serving as their carrier space. This carrier space leads to the concept of the vacuum state, which has no analog in the bra-ket formalism. Moreover, an essential difference is that the effect of second quantized operators depends on the occupancies of the one-electron levels in the wave function, since no annihilation is possible from an empty level and no electron can be created on an occupied spinorbital. At the same time, the occupancies of orbitals play no role in evaluating bra and ket expressions. Of course, both formalisms yield identical results after calculating the values of matrix elements. 

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