Algebraic quantization

Fried, L. E., and Ezra, G. S. (1988), PERTURB A Program for Calculating Vibrational Energies by Generalized Algebraic Quantization, Comp. Phys. Comm. 51, 103. 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

Nonautonomous Cases Perturhation Theory Based on Lie Transforms A simple Illustration of Algebraic Quantization LCPT with One Imaginary Frequency Mode... 

For a wide class of Hamiltonians described in Section II.B, a quite efficient technique, called algebraic quantization (AQ), was developed [38, 48-50]. This method hrst formally transforms if(p, q) to if(a, a) in terms of (a, a), which may correspond to customary creation and annihilation operators in quantum held theory, that is. 

The MFT equation of motion (25) can be derived in many ways, including the WKB approximation, the eikonal method, a (semi)classical time-dependent self-consistent field ansatz, density-matrix approaches, and the classical limit of algebraic quantization. Depending on the specific approach used, slightly different MFT schemes may result. For example, the classical force can be described either by the average of the quantum force as in Eq. (25) or by the derivative of the average quantum potential. 

In tetratomic molecules,1 there are three independent vector coordinates, iq, r2, and r3, which we can think of as three bonds. The general algebraic theory tells us that a quantization of these coordinates (and associated momenta) leads to the algebra... 

For molecules with many atoms, the simultaneous treatment of rotations and vibrations in terms of vector coordinates r1,r2,r3,..., quantized through the algebra... 

The algebraic treatment of polyatomic molecules proceeds in the same way as described previously. Each one-dimensional degree of freedom is quantized with the algebra of U(2),... 

Having established the correspondence between the Poschl-Teller potential and the algebra U(2), one can proceed to a quantization of bending vibrations along the lines of Section 4.2. We emphasize once more that the quantization scheme of bending vibrations in U(2) is rather different from that in U(4) and implies a complete separation between rotations and vibrations. If this separation applies, one can quantize each bending oscillator i by means of an algebra U,(2) as in Eq. (6.6). The Poschl-Teller Hamiltonian... 

Fano, U., and Racah, G. (1959), Irreducible Tensorial Sets, Academic Press, N.Y. Farrelly, D. (1986), Lie Algebraic Approach to Quantization of Nonseparable Systems with Internal Nonlinear Resonance, J. Chem. Phys. 85, 2119. 

The connected stmcture of the CSE has also been explored by Yasuda [23] using Grassmann algebra, by Kutzelnigg and Mukheijee [27] using a cumulant version of second-quantized operators, and by Herbert and Harriman [30] using a diagrammatic technique. 

Remark. It should be clear that this transition to an occupation number description is a purely algebraic step. In this respect it is similar to what in quantum mechanics is denoted by the misleading term second quantization . The only difference is that here we deliberately eliminate the information about the identity of the molecules , whereas in quantum mechanical applications (e.g., to photons or... 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

In Chapter 15, for the CFP with a detached electrons, we obtained a relationship (15.27) whose right side has the form of a vacuum average of a certain product of second-quantized operators q>. To obtain algebraic formulas for CFP, it is necessary to compute this vacuum average by transposing all the annihilation operators to the right side of the creation operators. So, for N = 3, we take into account (for non-repeating terms) the explicit form of operators (15.2) and (15.5), which produce pertinent wave functions out of vacuum, and find (cf. [107])... 

Similar expressions can also be obtained at N = 4 [108]. To establish similar algebraic expressions for the repeating terms vLS only requires that the appropriate second-quantized operators (16.7), (16.8), (16.10) and (16.11) be used. For example, instead of (16.63), we have... 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

In quantized theory, this is an operator in the fermion field algebra. Assuming mo = 0, the mean value (0 Af 0) vanishes in the reference vacuum state because all momenta and currents cancel out. In a single-electron state a) = al 0), a self-energy (more precisely, self-mass) is defined by Smc2 = a Mc2 a) = a / d3x y0(—eji)i/ a). Only helicity-breaking virtual transitions can contribute to this electromagnetic self-mass. 

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