Polarization moments, quantum

The fluorophore should be well characterized in terms of absorption and emission transition moments, quantum yield, polarization bands of interest, and behavior at different temperatures. The quantum yield should be high enough so that the level of probe needed for acceptably low signal noise would not be great enough to cause significant perturbation effects. 

The intermolecular forces of adhesion and cohesion can be loosely classified into three categories quantum mechanical forces, pure electrostatic forces, and polarization forces. Quantum mechanical forces give rise both to covalent bonding and to the exchange interactions that balance tile attractive forces when matter is compressed to the point where outer electron orbits interpenetrate. Pure electrostatic interactions include Coulomb forces between charged ions, permanent dipoles, and quadrupoles. Polarization forces arise from the dipole moments induced in atoms and molecules by the electric fields of nearby charges and other permanent and induced dipoles. 

The multipole (or polarization) moments introduced according to (2.14) present a classical analogue of quantum mechanical polarization moments [6, 73, 96,133, 304]. They are obtained by expanding the quantum density matrix [73, 139] over irreducible tensor operators [136, 140, 379] and will be discussed in Chapters 3 and 5. 

The quantitative characteristic of the alignment created is given, as already stated, by multipole moments of even rank. A more rigorous treatment of the expansion of the quantum mechanical density matrix over irreducible tensorial operators will be performed later, in Chapter 5 and in Appendix D. As an example we will write the zero, second and fourth rank polarization moments and [Pg.62]

We will first try to understand the basic outlines of the phenomena on the basis of the framework of polarization moments, as treated in the preceding chapters. In order to avoid overloading the text with excessive formalism and in order to achieve easier understanding, we will consider a simplified model in the present chapter which gives an idea of the essence of the phenomena. The possibility of a more comprehensive quantum mechanical description will be offered by the equations presented in the following chapter. 

Other forms of normalization, as well as forms denoting irreducible tensor operators may be found in [304] and in Appendix D. With the aid of the orthogonality relation one may easily express the quantum mechanical polarization moments fq and Pq through the elements of the density matrix /mm and... 

Since the density matrix is Hermitian, we obtain the property of polarization moments which is analogous to the classical relation (2.15) fq = (—1 ) (f-q) and tp = (—l) 3( g). The adopted normalization of the tensor operators (5.19) yields the most lucid physical meaning of quantum mechanical polarization moments fq and p% which coincides, with accuracy up to a normalizing coefficient that is equal for polarization moments of all ranks, with the physical meaning of classical polarization moments pq, as discussed in Chapter 2. For a comparison between classical and quantum mechanical polarization moments of the lower ranks see Table 5.1. 

Table 5.1. Comparison between classical pq and quantum mechanical fq polarization moments...

Summing up the above, we may conclude that the classical system of equations (5.93) and (5.94), together with the above given additional terms, coincides perfectly with the asymptotic system of equations of motion of quantum mechanical polarization moments (5.87) and (5.88). This result was actually to be expected from correspondence principle considerations. 

In order to analyze the electric field induced level splitting effect we will again expand /mm over quantum polarization moments fq. As may be seen from (5.40), which is the reverse of (6.21), we obtain in the x, y, z frame... 

Appendix D. Various methods of introducing polarization moments D.l Polarization moments in quantum physics... 

In the present book we have used the cogredient expansion form (2.14), where, as distinct from the standard form, an additional normalizing factor has been introduced, namely (—l)< v/(2K + l)/4n. Our expansion of the classical probability density p(0, differs from the standard one in exactly the same way as the expansion of the quantum mechanical density matrix p over 2Tq differs from the expansion over lTg. In Section 5.3 we present a comparison between the physical meaning of the classical polarization moments pg, as used in the present book, and the quantum mechanical polarization moments fg, as determined by the cogredient method using normalization (D.ll). 

Thus, we have attempted to give, in the present appendix, an idea of the various methods of determining classical and quantum mechanical polarization moments and some related coefficients. We have considered only those methods which are most frequently used in atomic, molecular and chemical physics. An analysis of a great variety of different approaches creates the impression that sometimes the authors of one or other investigation find it easier to introduce new definitions of their own multipole moments, rather than find a way in the rather muddled system of previously used ones. This situation complicates comparison between the results obtained by various authors considerably. We hope that the material contained in the present appendix might, to some extent, simplify such a comparison. 

In conclusion, we have presented a reliable method for selective production and detection of high-order atomic polarization moments based on the nonlinear magneto-optical effects with frequency modulated light. This method can be used for the selective control of higher order atomic coherences in multilevel systems exploiting large Kerr nonlinearities for the construction of all-optical quantum phase gates. 

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. 

Hartree-Fock, DFT or CCSD levels. Because they can reproduce such quantities, APMM procedures should account for an accurate description of the interactions including polarization cooperative effects and charge transfer. They should also enable the reproduction of local electrostatic properties such as dipole moments an also facilitate hybrid Quantum Mechanical/Molecular Mechanical (QM/MM) embeddings. 

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