Quantum mechanics definition

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 ... 

The expression for J is derived via the general quantum mechanical definition (32), introducing the perturbation expansion for the current density and the a-state molecular wave-function (depending on n-electron space-spin coordinates ), yb—Zd e-... 

The quantum mechanical definition of a distribution function in the classical phase-space is an old theme in theoretical physics. Most frequently used is the so-called Wigner distribution function (Wigner 1932 Hillery, O Connell, Scully, and Wigner 1984). Let us consider a onedimensional system with coordinate R and corresponding classical momentum P. The Wigner distribution function is defined as... 

Despite the success in parameterizing acid/base and many other properties for a range of different compounds, it is obvious that the simple electrostatic model used by Taft and extended by others [27, 28] have fundamental weaknesses - both with regards to the domain of validity and at a more fundamental level. The model is more intuitive than physical, in the sense that the inductive effect, polarisation effect, resonance effect, mesomeric effect and steric effect have no proper quantum mechanical definition, and can therefore not be derived directly from the system s wave function [29,30]. 

Problems associated with the quantum-mechanical definition of molecular shape do not diminish the importance of molecular conformation as a chemically meaningful concept. To find the balanced perspective it is necessary to know that the same wave function that describes an isolated molecule, also describes the chemically equivalent molecule, closely confined. The distinction arises from different sets of boundary conditions. The spherically symmetrical solutions of the free molecule are no longer physically acceptable solutions for the confined molecule. 

This interesting field was initiated by Bader [158]. Topological analysis provides the means for a concise description of multivariate functions. For functions that describe physical observables, the number and location of critical points, where the gradient vanishes, and their mutual relationship are often directly related to the properties of the system under study. The application of topological analysis to the one electron density is even more productive, furnishing rigorous quantum-mechanical definitions of and bonds in molecules. Cioslowski has extended this analysis to the study of the electron-electron interactions, based on the analysis of the intracule and extracula densities [159,160]. 

TTie definition of a bound atom—an atom in a molecule— must be such that it enables one to define all of its average properties. For reasons of physical continuity, the definition of these properties must reduce to the quantum mechanical definitions of the corresponding properties for an isolated atom. The atomic values for a given property should, when summed over aU the atoms in a molecule, yield the molecular average for that property The atomic properties must be additive in the above sense to account for the observation that, in certain series of molecules, the atoms and their properties are transferable between molecules, leading to what are known as additivity schemes. An additivity scheme requires both that the property be additive over the atoms in a molecule and that the atoms be essentially transferable between molecules. 

In the LCAO MO description, the H2 molecnle in its ground state has a pair of electrons in a bonding MO, and thus a single bond (that is, its bond order is 1). Later in this chapter, as we describe more complex diatomic molecules in the LCAO approximation, bond orders greater than 1 are discussed. This quantum mechanical definition of bond order generalizes the concept first developed in the Lewis theory of chemical bonding—a shared pair of electrons corresponds to a single bond, two shared pairs to a double bond, and so forth. 

As the labels imply, the f s provide the geometric definition of the four half body-diagonals of a cube, which define a tetrahedron. When the definitions of 9 are compared with Equation (2) the symbols p and Py are seen to be the identification of Cartesian axes in polar coordinates. The sp linear combination is thereby identified as a geometrical, rather than a quantum-mechanical, definition of the four assumed tetrahedral bonds of methane in polar coordinates. 

In this expression is the electric dipole polarizability, is the mixed electric dipole-quadrupole polarizability, Xap is th magnetic susceptibility [56], rj p y and rj p ys can, respectively, be called electric dipole and electric quadrupole polarizability of the magnetic susceptibility. Quantum-mechanical definitions for these quantities within the framework of the Rayleigh-Schrddinger perturbation theory are given later. 

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. 

Now consider the quantum-mechanical definition of the electric dipole moment. Suppose we apply a uniform external electric field E to an atom or molecule and ask for the effect on the energy of the system. To form the Hamiltonian operator, we first... 

This correspondence is hardly surprising since the quantum mechanical definition of angular momentum is, say, for Lx... 

Now consider the quantum-mechanical definition of the electric dipole moment. Suppose we apply a uniform external electric field E to an atom or molecule and ask for the effect on the energy of the system. To form the Hamiltonian operator, we first need the classical expression for the energy. The electric field strength E is defined as E = F/Q, where F is the force the field exerts on a charge Q. We take the z direction as the direction of the applied field E = The potential energy V is [Eq. (4.24)]... 

The beauty of the above topological definition of the atom in a molecule lies in the fact that it coincides with the rigorous quantum mechanical definition of an open subsystem [27, 33, 34]. In particular, the atomic action integral, which is defined through the atomic one-particle Lagrangian density, is zero within the atomic volume ... 

The Hamiltonian operator acts on to give the permitted energy levels for the molecule. But what is In the quantum-mechanical definition, is a function that contains all the information that is possible to know about a system. This information can be obtained by acting on with the appropriate operator, e.g. the Hamiltonian, to recover the total energy. 

The central aim of this chapter is to give a simple, self-contained approach to a set of molecular magnetic properties in terms of induced current densities and related property density maps, via classical relationships combined with quantum mechanical definitions, and computational procedures. Some efforts are made to document the effectiveness of such a theoretical treatment, in the attempt to rationalize the phenomenology and to form a mental image of the mechanisms underlying the electronic interaction with static magnetic perturbations. 

Perturbation theory has been used to define molecular magnetic properties up to fourth order. The invariance of the response tensors in a gauge transformation of the vector potential has been analyzed, and its connections with the conditions for charge and current conservations have been discussed. The quantum mechanical definition of electron current density in the presence of static electric and magnetic fields has been employed to provide relationships for magnetic... 

Solutions of equations of the type (3.27) are not straigfatfduward. First, all equilibrium bond moment values must be known. In the genmal case, a correct quantum mechanical definition of a "bond moment" is difficult to produce, if possible at all. Attempts to define bcxid moments quantum-mechanicaUy are always based on severe approximations [79]. 

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