Harmonic oscillator information theoretical

These are all empirical measurements, so the model of the harmonic oscillator, which is pur ely theoretical, becomes semiempirical when experimental information is put into it to see how it compares with molecular vibration as determined spectroscopically. In what follows, we shall refer to empirical molecular models such as MM, which draw heavily on empirical information, ab initio molecular models such as advanced MO calculations, which one strives to derive purely from theory without any infusion of empirical data, and semiempirical models such as PM3, which are in between (see later chapters). 

The theoretical background which will be needed to calculate the excited state distortions from electronic and Raman spectra is discussed in this section. We will use the time-dependent theory because it provides both a powerful quantitative calculational method and an intuitive physical picture [42,46-50]. The method shows in a simple way the inter-relationship between Raman and electronic spectroscopy. It demonstrates that the intensity of a peak in a resonance Raman spectrum provides detailed information about the displacement of the excited state potential surface along the normal mode giving rise to the peak [42,48]. It can also be used to calculate distortions from the intensities of vibronic peaks in electronic spectra [49]. For harmonic oscillators, the time-dependent theory is mathematically equivalent to the familiar Franck-Condon calculation [48]. 

At T = 0 all the particles reside in the lowest state Go= (3/2) oc)ho- The critical temperature [Eq. (9)] depends on the total number of particles (with a finite value of cohoA ) and not on the density, as is the case for the homogeneous system [Eq. (6)]. The different temperature dependence for the condensate fraction for the confined boson gas [Eq. (10)] and for the uniform Bose gas [Eq. (8)] can be traced to the higher density of states for the harmonic oscillator relative to that for a particle in a box [14, 24]. Theoretical studies for finite size effects in an ideal finite Bose gas [80, 126] and for a Bose gas trapped in a harmonic potential [14, 127] provided novel information on finite boson systems. These issues will be addressed in Section I.E. 

In this section, we present some general results on the information theoretical uncertainty-like measures applicable to the standard model systems of hydrogen-like atoms and the isotropic harmonic oscillator. The characteristic features of the spherically confined systems will be highlighted. 

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