Harmonic oscillator vibrational states

The harmonic-oscillator stationary-state energy levels (4.47) are equally spaced (Fig. 4.1). Do not confuse the quantum number v (vee) with the vibrational frequency V (nu). 

Evaluate fm H f ) if (a) H is the harmonic-oscillator Hamiltonian operator and f and f are harmonic-oscillator stationary-state wave functions with vibrational quantum numbers m and n (b) H is the particle-in-a-box H and / and f are particle-in-a-box energy eigenfunctions with quantum numbers m and n. 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

The vibrational and rotational motions of the chemically bound constituents of matter have frequencies in the IR region. Industrial IR spectroscopy is concerned primarily with molecular vibrations, as transitions between individual rotational states can be measured only in IR spectra of small molecules in the gas phase. Rotational - vibrational transitions are analysed by quantum mechanics. To a first approximation, the vibrational frequency of a bond in the mid-IR can be treated as a simple harmonic oscillator by the following equation ... 

The reason that AG/ = Ef — ) is due to the fact that our system consists of a collection of harmonic oscillators with displaced surfaces between the initial and final electronic states. In this case, the vibrational partition functions are the same between the initial and final electronic states. 

The book thus embraces an extended study on a variety of issues within the theory of orientational ordering and phase transitions in two-dimensional systems as well as the theory of anharmonic vibrations in low-dimensional crystals and dynamic subsystems interacting with a phonon thermostat. For the sake of readability, the main theoretical approaches involved are either presented in separate sections of the corresponding chapters or thoroughly scrutinized in appendices. The latter contain the basic formulae of the theory of local and resonance states for a system of bound harmonic oscillators (Appendix 1), the theory of thermally activated reorientations and tunnel relaxation of orientational... 

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