Wave function well-behaved, defined

Finally, we plot in Fig. 3 the ellipticities e i = iJLi(m i )liJL2im i ) and 2 = Hm2( 2)/mi( 2) of the waves corresponding to mi and m2 as a function of reduced wavelength X. These have been defined so that linear polarization is associated with a value of zero one could equally well have chosen to plot the reciprocal relations in which case linear polarization would correspond to infinite ellipticity. With either definition, the ellipticity of circularly polarized waves is unity. It is apparent that the ellipticity of the m 1 wave is anomalous in the region of the reflection band whereas the m2 wave is well behaved for all values of X. ... 

The metallic bond can be seen as a collection of molecular orbitals between a large number of atoms. As Figure A.6 illustrates, the molecular orbitals are very close and form an almost continuous band of levels. It is impossible to detect that the levels are actually separated from each other. Rather, the bands behave in many respects similarly to the orbitals of the molecule in Figure A. 5 if there is little overlap between the electrons, the interaction is weak and the band is narrow. This is the case for the d-electrons of the metal. Atomic d-orbitals have pronounced shapes and orientations that are largely retained in the metal. This in contrast to the s electrons, which are strongly delocalized that is, they are not restricted to well-defined regions between atoms, and form an almost free electron gas that spreads out over the whole metal. Hence, the atomic s-electron wave functions overlap to a great extent, and consequently the band they form is much broader. 

Any well-behaved periodic function (such as a spectrum) can be represented by a Fourier series of sine and cosine waves of varying amplitudes and harmonically related frequencies. A typical NIR spectrum may be defined mathematically by a series of sines and cosines in the following equation ... 

A complete set of functions of a given variable has the property that a linear combination of its members can be used to construct any well-behaved function of that variable, even if the target function is not a member of the set. Well-behaved in this context means a function that is finite everywhere in a defined interval and has finite first and second derivatives everywhere in this region. Fourier analysis uses this property of the set of sine and cosine functions (Appendix A.3). In quantum mechanics, wavefunctions of complex systems often are approximated by combinations of the wavefunctions of simpler or ideahzed systems. Wave-functions of time-dependent systems can be described similarly by combining wavefunctions of stationary systems in proportions that vary with time. We ll return to this point in Sects. 2.3.6 and 2.5. 

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