Polarization moments, quantum physical meaning

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. 

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). 

In this chapter, we extend our treatment of rotation in diatomic molecules to nonlinear polyatomic molecules. A traditional motivation for treating polyatomic rotations quantum mechanically is that they form a basis for experimental determination for bond lengths and bond angles in gas-phase molecules. Microwave spectroscopy, a prolific area in chemical physics since 1946, has provided the most accurate available equilibrium geometries for many polar molecules. A background in polyatomic rotations is also a prerequisite for understanding rotational fine structure in polyatomic vibrational spectra (Chapter 6). The shapes of rotational contours (i.e., unresolved rotational fine structure) in polyatomic electronic band spectra are sensitive to the relative orientations of the principal rotational axes and the electronic transition moment (Chapter 7). Rotational contour analysis has thus provided an invaluable means of assigning symmetries to the electronic states involved in such spectra. 

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