Statistics, quantum-mechanical

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

The topic that is commonly referred to as statistical quantum mechanics deals with mixed ensembles only, although pure ensembles may be represented in the same formalism. There is an interesting difference with classical statistics arising here In classical mechanics maximum information about all subsystems is obtained as soon as maximum information about the total system is available. This statement is no longer valid in quantum mechanics. It may happen that the total system is represented by a pure ensemble and a subsystem thereof by a mixed ensemble. 

The conclusion is that the prescriptions of statistical quantum mechanics, e.g., that governing the way a thermal state is defined in Eq. (42), cannot explain chemical phenomena without taking over concepts from traditional chemistry in an ad hoc manner. These prescriptions do not give rise to (i) molecular isomers, (ii) handed molecules, (iii) monomer sequences in a macromolecule, or (iv) differently knotted macromolecules. For all these chemically well-known concepts, different expectation values of the nuclear position operators are necessary. 

In summary, the usual prescriptions of statistical quantum mechanics cannot explain chemical phenomena such as isomerism or the handedness of molecules. The question is then how to introduce effective thermal states for different isomers or differently handed molecules, etc. 

In summary, statistical quantum mechanics permits us to derive strictly classical observables (such as the classical specific magnetization operator) by appropriate limit considerations (such as a limit of infinitely many spins in case of the Curie-Weiss model). However, statistical quantum mechanics cannot cope with fuzzy classical observables (for finitely many degrees of freedom) since different decompositions of a thermal state Dp are considered to be equivalent. The introduction of a canonical decomposition of Dp into pure states will give rise to an individual formalism of quantum mechanics in which fuzzy classical observables can be treated in a natural way. 

Thermal states, as defined by the usual prescription of statistical quantum mechanics [see Eq. (42)] are in contradiction with traditional chemistry. Neither isomers, nor handed states, nor any other structural chemical concepts can be described or explained thereby. 

Hence, starting from usual statistical quantum mechanics, we find it impossible to understand traditional chemical theories about single molecules. 

In the following three sections we shall discuss four applications of quantum mechanics to miscellaneous problems, selected from the very large number of applications which have been made. These are the van der Waals attraction between molecules (Sec. 47), the symmetry properties of molecular wave functions (Sec. 48), statistical quantum mechanics, including the theory of the dielectric constant of a diatomic dipole gas (Sec. 49), and the energy of activation of chemical reactions (Sec. 50). With reluctance we omit mention of many other important applications, such as to the theories of the radioactive decomposition of nuclei, the structure of metals, the diffraction of electrons by gas molecules and crystals, electrode reactions in electrolysis, and heterogeneous catalysis. 

STATISTICAL QUANTUM MECHANICS. SYSTEMS IN THERMODYNAMIC EQUILIBRIUM... 

The subject of statistical mechanics is a branch of mechanics which has been found very useful in the discussion of the properties of complicated systems, such as a gas. In the following sections we shall give a brief discussion of the fundamental theorem of statistical quantum mechanics (Sec. 49a), its application to a simple system (Sec. 496), the Boltzmann distribution law (Sec. 49c), Fermi-Dirac and Bose-Einstein statistics (Sec. 49d), the rotational and vibrational energy of molecules (Sec. 49e), and the dielectric constant of a diatomic dipole gas (Sec. 49/). The discussion in these sections is mainly descriptive and elementary we have made no effort to carry through the difficult derivations or to enter into the refined arguments needed in a... 

A new statistical quantum mechanical approach - the statistical adiabatic product distribution method has been applied to the photodissociation of ketene in parallel theoretical and experimental studies. The main focus of the work, however, was H-atom production via C-H cleavage rather than the elimination of CO. The photodecomposition of formohydroxamic acid (HCONHOH) has been investigated in matrix-isolation FTIR and DFT stud-ies. " Irradiation of the acid in Ar or Xe matrices with the full output of a xenon arc lamp generated H-bonded HNCO- H2O and NH20H- CO complexes. In the latter case, the IR spectra also suggest the existence of a structure with the NH2 group interacting with the carbon atom. 

---