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8 - Mechanical Vibrations and the Harmonic Oscillator Model

Published online by Cambridge University Press:  11 May 2023

Uri Peskin
Affiliation:
Technion - Israel Institute of Technology, Haifa
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Summary

Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.

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Publisher: Cambridge University Press
Print publication year: 2023

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References

Abramowitz, M. and Irene, A. S., eds., “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.” National Bureau of Standards Applied Mathematics Series, vol. 55. (US Government Printing Office, 1964).Google Scholar
Martin, J. D. D. and Hepburn, J. W., “Determination of bond dissociation energies by threshold ion-pair production spectroscopy: An improved D (HCl).” The Journal of Chemical Physics 109, 8139 (1998).CrossRefGoogle Scholar

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