Angular momentum theory

This book presents a detailed exposition of angular momentum theory in quantum mechanics, with numerous applications and problems in chemical physics. Of particular relevance to the present section is an elegant and clear discussion of molecular wavefiinctions and the detennination of populations and moments of the rotational state distributions from polarized laser fluorescence excitation experiments. 

A nucleus having spin generates a magnetic moment pi. which is proportional to the angular momentum. Theory is not capable of calculating pi, so it is commonly expressed as Eq. (4-42), where 7 is called the magnetogyric ratio. 

This brief discussion of the physical meaning and mutual correspondence of different models and theories of rotational motion is intended as a guide for those who do not intend to examine the book systematically. Setting forth the material consistently, one cannot avoid certain formalisms peculiar to angular momentum theories. We hope, however, that a detailed commentary will enable readers to form a clear notion of the most important assumptions and results without referring to proofs. 

Several theories have been developed to explain the rainbow phenomena, including the Lorenz-Mie theory, Airy s theory, the complex angular momentum theory that provides an approximation to the Lorenz-Mie theory, and the theory based on Huy gen s principle. Among these theories, only the Lorenz-Mie theory provides an exact solution for the scattering of electromagnetic waves by a spherical particle. The implementation of the rainbow thermometry for droplet temperature measurement necessitates two functional relationships. One relates the rainbow angle to the droplet refractive index and size, and the other describes the dependence of the refractive index on temperature of the liquid of interest. The former can be calculated on the basis of the Lorenz-Mie theory, whereas the latter may be either found in reference handbooks/literature or calibrated in laboratory. 

Judd, B. R. (1975), Angular Momentum Theory for Diatomic Molecules, Academic Press, New York. 

Judd, B.R., Angular Momentum Theory for Diatomic Molecules, Academic Press, New York, (1975). Monkhorst, H. and Jeziorski, B., J. Chem. Phys. 71, 5268 (1979). 

Vol. 47 C.A. Morrison, Angular Momentum Theory Applied to Interactions in Solids. 8,9-159 pages. 1988. 

In order to utilize the mathematical apparatus of the angular momentum theory, we have to write all the operators in. /-representation, i.e., to express them in terms of the quantities which transform themselves like the eigenfunctions of operators j2 and jz. For example, the explicit form of the spherical components of the spin operator in. /-representation, according to [28], is as follows ... 

Foundations of the Angular Momentum Theory. Graphical Methods... 

In angular momentum theory a very important role is played by the invariants obtained while summing the products of the Wigner (or Clebsch-Gordan) coefficients over all projection parameters. Such quantities are called 7-coefficients or 3ny-coefficients. They are invariant under rotations of the coordinate system. A j-coefficient has 3n parameters (n = 1,2,3,...), that is why the notation 3nj-coefficient is widely used. The value n = 1 leads to the trivial case of the triangular condition abc, defined in Chapter 5 after formula (5.25). For n = 2,3,4,... we have 67 -, 9j-, 12j-,. .. coefficients, respectively. 3nj-coefficients (n > 2) may be also defined as sums of 67-coefficients. There are also algebraic expressions for 3nj-coefficients. Thus, 6j-coefficient may be defined by the formula... 

The non-relativistic wave function (1.14) or its relativistic analogue (2.15), corresponds to a one-electron system. Having in mind the elements of the angular momentum theory and of irreducible tensors, described in Part 2, we are ready to start constructing the wave functions of many-electron configurations. Let us consider a shell of equivalent electrons. As we shall see later on, the pecularities of the spectra of atoms and ions are conditioned by the structure of their electronic shells, and by the relative role of existing intra-atomic interactions. 

The mathematical apparatus of the angular momentum theory can be applied to describe the tensorial properties of electron creation and annihilation operators in the space of occupation numbers of a certain definite one-particle state a). It follows from (13.29) and (13.30) that the operators... 

The treatment in this book relies on the techniques of angular momentum theory. The author has sought, as far as possible, to do without higher-rank groups. Therefore, we shall only give here a glimpse of those methods, so as to be able to show later on how and to what degree they... 

A. P. Jucys, I. B. Levinson and V. V. Vanagas. Mathematical Apparatus of the Angular Momentum Theory, Vilnius, 1960 (in Russian). English translations Israel Program for Scientific Translations, Jerusalem 1962 Gordon and Breach, New York, 1963. 

The development of a pseudo-angular momentum theory of (t2g)n states, which is rooted in the point group under consideration can thus proceed in the following way one first converts the T2 basis into a T1 basis. The 7 functions can then be coupled to multiplet states using the coupling theory of the full... 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

The use of quarks in atomic shell theory provides an alternative basis to the traditional one. The transformations between these bases can be complicated, but there are many special cases where our quarks can account for unusual selection rules and proportionalities between sets of matrix elements that, when calculated by traditional methods, go beyond what would be predicted from the Wigner-Eckart theorem [4,5], This is particularly true of the atomic f shell. An additional advantage is that fewer phase choices have to be made if the quarks are coupled by the standard methods of angular-momentum theory, for which the phase convention is well established. This is a strong point in favor of quark models when icosahedral systems are considered. A number of different sets of icosahedral Clebsch-Gordan (CG) coefficients have been introduced [6,7], and the implications of the different phases have to be assessed when the CG coefficients are put to use. 

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