The Quantized Harmonic Oscillator Vibrational Spectroscopy

The previous particle-in-a-box (PIB) and particle-on-a-ring (POR) problems both had V=0 and only dealt with the kinetic energy operator. The essence of the harmonic oscillator is a parabolic kjp  

This could have been solved to obtain the same result in more complicated ways but we want to buUd on your experience from problems in physics and we note that A = amplimde.  

FIGURE 12.1 A parabolic potential superimposed on a Morse potential using actual data for the H2 molecule. An exact but more complicated solution also exists for the Morse potential. (See Ref. [3]) 

The graph shows that the harmonic and Morse potential wells superimpose fairly well at low energy and probably the first few energy levels correspond for both models. That may be all we need for ground states most of the time on planet Earth. The exact potential well can also be constructed from spectroscopic data but it is adequate to introduce the Morse potential to provide a useful model. Both the parabolic and Morse potentials are incorrect in that the short range part crosses the vertical axis which is meaningless so the potentials are only shown for x 0 but the lit for low energy levels is still useful. 

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Since the very beginning, almost a century ago, spectroscopy has revealed that vibrational dynamics are quantal in nature the energy is quantized and dynamics are described in terms of eigenstates. The quantum theory has developed predominantly within the harmonic approximation, that is the simplest expansion, to quadratic terms, of the potential hypersurface around the equilibrium position. Vibrational dynamics can be thus represented with harmonic oscillators (normal modes) corresponding to coherent oscillations of all degrees... 

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