Quantum object definition

If population analysis is not synonymous with the concept of an AIM, it becomes necessary to introduce a proper set of requirements before one can speak of an AIM. An AIM is a quantum object and as such has an electron density of its own. This atomic electron density must obviously be positive definite and the sum of these atomic densities must equal the molecular density. Each atomic density pA(r) can be obtained from the molecular density p(r) in the following way ... 

The most interesting example of a quantum mechanical object is the photon itself. By using the relativistic and quantum mechanical definition of the photon energy, we can obtain a quantitative formulation of the concepts just described. The relativistic form of the total energy of a particle with rest mass m and momentum p is ... 

Once a set of quantum objects to study is chosen and the operator related to the MQSM definition in Eq. (2) is defined, the MQSM related to the set is unique. But they can be transformed or combined in order to obtain a new kind of auxiliary terms which can be named Quantum Similarity Indices (QSI). A vast quantity of possible MQSM manipulations leading to a variety of QSI definitions exists. The most used QSI are as follows. 

The mean radius of a Rydberg state is approximately aon 2, where ao is the Bohr radius. Although the atom in its ground state is a quantum object, it eventually ceases to be so as n increases for n 100, the diameter of the atom is 104 x 2ao which is roughly 1 /xm, at which point the atom is definitely reaching macroscopic size. For n 1000, the diameter of the atom is l/10th of a millimetre. 

Density Functions play a fundamental role in the definition of Quantum Theory, due to this they are the basic materials used in order to define Quantum Objects and from this intermediate step, they constitute the support of Quantum Similarity Measures. Here, the connection of Wavefunctions with Extended Density Functions is analysed. Various products of this preliminary discussion are described, among others the concept of Kinetic Energy Distributions. Another discussed set of concepts, directly related with the present paper, is constituted by the Extended Hilbert Space definition, where their vectors are defined as column structures or diagonal matrices, containing both wavefunctions and their gradients. The shapes of new density distributions are described and analysed. All the steps above summarised are completed and illustrated, when possible, with practical application examples and visualisation pictures. 

Several new possibilities have become apparent along the path of the QSM theoretical development [44], Among others, one can quote Electrostatic Potential distributions as well as eDF transformations. Thus, one can say that eDF analysis and the attached concept of Quantum Object (QO) [41,45,46] have opened the way towards an almost complete definition of the structure of QSM and their generalisations. At the end, a mathematical picture of the connection between chemical information and the idea of molecule emerges in the form of Tagged Sets and Ensembles [47,48], see also the Definition 1 and Definition 8 of the Appendix A. 

The idea of Quantum Object (QO) without a well-designed formulation has been already used in the literature [30-32,35]. Moreover, the background mathematical structure leading towards the recently published [48,49] definition of QO is to be found within the possibility to construct a new kind of sets the Tagged Sets. In their definition, both set elements and known information on them are taken simultaneously into account. In order to obtain a sound QO definition, besides the definition of the Tagged Set concept [47], and furthermore to develop this line of thought, there are needed some preliminary considerations. 

One rather radical assumption has had to be made that, namely, of the indistinguishability of molecules, which converted the Boltzmann definition of the entropy into the quantum mechanical definition, and proved essential for the calculation of the absolute entropy. This represents the most drastic departure which we have so far met from the naive conception of molecules as small-scale reproductions of the recognizable macroscopic objects around us. But still more drastic departures will prove necessary. 

The quantum mechanical viable definition being an intrinsic property of an atom, that is a quantum object, the electronegativity has to include the quantum nature of the electronic systems to whom is associated. The present scales are accommodated within conceptual DFTby employing the softness realization (4.226), which contains three quantum constraints such the translational invariance condition, the Hellmann-Feynman theorem, and the normalization of the linear response function are. 

The idea of molecular quantum similarity can be extended to other operators, provided they are positive definite. In this sense, they will lead to real, positive definite values for the MQSM evaluated over the density functions of the involved quantum objects. 

A similarity measure over the unit shell of any VSS can be defined through the description of the mathematical elements, which have been described so far. In the simplest way, an MQSM can be defined knowing the appropriate density function tags of two quantum objects Pa Pb j adapted to some shells of the corresponding VSS, and with as a weight some positive definite operator ff. In that case, the integral measure... 

There is no consensus about how to define atomic sizes. As a quantum object, the atom has no clear-cut boundary and no definite size electron density of an isolated atom drops to zero only at infinite distance. However, nearly all this electron density... 

MQOS are a particular class of tagged sets. MQOS are molecular sets, where each element is described via a quantum mechanical density function. The set of molecular structures is termed the object set and the set of attached density functions is the tag set. The couples (submicroscopic element density function tag) are termed quantum objects. See below in the section The mathematical structure of the collection cf TDQS integrals, when computed over a molecular quantum object set (MQOS), for further detailed definitions and references. See also reference Ref. [22] for an up to date account of terms, references and definitions. 

The Cartesian product of both sets O = M x P is called a molecular quantum object set (MQOS) [22, 52-54]. This MQOS definition constitutes a way to put together and resume into a unique structure the Eq. (8). 

A vector semispace is considered here as a vector space defined over the positive real numbers. In this way the additive group of the semispace is a semigroup, a group without reciprocal elements. DF sets like the quantum object tags already discussed in MQOS definition are subsets of some Hilbert semispace. 

The three quantum numbers may be said to control the size (n), shape (/), and orientation (m) of the orbital tfw Most important for orbital visualization are the angular shapes labeled by the azimuthal quantum number / s-type (spherical, / = 0), p-type ( dumbbell, / = 1), d-type ( cloverleaf, / = 2), and so forth. The shapes and orientations of basic s-type, p-type, and d-type hydrogenic orbitals are conventionally visualized as shown in Figs. 1.1 and 1.2. Figure 1.1 depicts a surface of each orbital, corresponding to a chosen electron density near the outer fringes of the orbital. However, a wave-like object intrinsically lacks any definite boundary, and surface plots obviously cannot depict the interesting variations of orbital amplitude under the surface. Such variations are better represented by radial or contour... 

Definition of the relativistic quantum rotor is given earlier (Aldinger et.al., 1983). According to this definition the relativistic quantum rotor is an object having three limits ... 

In this section, we provide a succinct survey of the definitions and results of group theory required for a group-theoretical formulation of quantum and classical mechanics, which are the objects of the following sections. 

The breadth in scope of this definition comes at the cost of requiring flexibility and judgment in the interpretation of object, superposable, and mirror image. The model must suit the occasion of its use. Chirality, with respect to an isolated molecule, is a quantum-mechanically undefined concept, but, because we... 

The main objection against the Bohr and Sommerfeld atomic models was the ad hoc definition of stationary states. Simply declaring these as quantum states offers no explanation for the failure of an accelerated charge to radiate energy. The quantization of neither energy nor angular momentum implies such an effect. 

When first confronted with the oddities of quantum effects Bohr formulated a correspondence principle to elucidate the status of quantum mechanics relative to the conventional mechanics of macroscopic systems. To many minds this idea suggested the existence of some classical/quantum limit. Such a limit between classical and relativistic mechanics is generally defined as the point where the velocity of an object v —> c, approaches the velocity of light. By analogy, a popular definition of the quantum limit is formulated as h —> 0. However, this is nonsense. Planck s constant is not variable. 

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