Quantization techniques

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,... 

This survey of methods for obtaining configuration state functions has necessarily been brief, since the topic could easily occupy a course of its own. However, we have treated the methods in common use, and much additional material on second quantization techniques and matrix element evaluation will be covered elsewhere. 

Moreover, the second-quantization technique is closely linked with the quasispin method, which turned out to be fruitful in atomic theory, since it enables physicists to improve significantly the theory of the spectra of many-electron atoms and ions, and to make it far more universal, and more simple and convenient for practical use. 

Thus, from these brief introductory remarks we can see that the second-quantization technique looks very promising in atomic spectroscopy. Therefore, let us discuss in this part all the above-mentioned questions in more detail. The discussion is mainly based on the monograph [31]. 

The concept of quasispin quantum number was discussed in the Introduction and Chapter 9 (see formulas (9.22) and (9.23)). Now let us consider it in the framework of the second-quantization technique. We can introduce the following bilinear combinations of creation and annihilation operators obeying commutation relations (14.2) - the quasispin operator ... 

In accordance with the secondary quantization technique, the operator if/ can be presented in the form ... 

Similar to the evaluation of the SOC matrix element, it is convenient to employ second quantization technique to compute SSC matrix elements over a wavefunction of a MRCI type. In the second quantized formulation, the RMEs of the SSC operator reads ... 

Another formulation of this model using the second quantization technique has also been proposed122). The basic ideas as well as the main conclusions are unchanged. 

Leich, H., Deketelaere, S., Dbman, L, Dothey, M., and Wery. A new quantization technique for LSP parameters and its application to low bit rate multi-band excited vocoders. In EUSIPCO 1992). 

A number of quantization techniques are available, and the selection of the optimum analog-to-digital converter (ADC) for a particnlar application is properly based on considerations of resolution (precision), accnracy (initial and drift with time and temperature), ease of interfacing, cost, and convenience (availability, size and power requirements). To select an ADC, it is usefnl to nnderstand exactly what types are available and how they work. The fisting in Table 3.1.1, althongh far from complete, does include the most popular ADCs, especially those currently prodnced in an integrated circuit form. 

The quark model is a mine of exercises for those who have the chance of teaching quantum mechanics, once they have exhausted the charm of the Stark effect and other examples borrowed from atomic physics. The baryon sector is particularly rich. For instance, it illustrates how antisymmetrization is important, in a situation intermediate between the trivial two-body case and the limit of a large number of constituents, where second-quantization techniques are applied. It is also amazing to show that, if qqq is bound by a pairwise potential, V= E whose strength is half of that of the qq potential, i.e., u(r) = V r), then the energies or masses of ground-state mesons and baryons fulfil the inequality 2(qqq) > 3(qq). This is a simple consequence of the variational principle, as shown in chapter 9. 

Beside the above three examples, there are many formulae which are much easier to obtain by the second quantized technique. The interested reader may exercise in trying to rederive some other well-known results of quantum chemistry. 

Schhck, C. Quantization techniques for visualization of high dynamic range pictures. In Photorealistic Rendering Techniques, pp. 7-20 (1995)... 

Also, it is possible to use semiclassical quantization techniques to select the vibrational states of triatomic molecules taking part in the chemical reactions. Good examples of QCT applications are the reactions... 

There are many excellent text books on second-quantization techniques to which we refer the interested reader for a more thorough discussion than that given here. We mention, in particular ... 

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