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9. Harmonic Oscillator - MIT OpenCourseWare

9. Harmonic Oscillator Harmonic Oscillator Classical Harmonic Oscillator and model Oscillator Hamiltonian: Position and momentum operators Position representation Heisenberg picture Schro dinger picture Uncertainty relationships Coherent States Expansion in terms of number states Non-Orthogonality Uncertainty relationships X-representation Phonons Harmonic Oscillator model for a crystal Phonons as normal modes of the lattice vibration Thermal energy density and Specific Heat Harmonic Oscillator We have considered up to this moment only systems with a finite number of energy levels; we are now going to consider a system with an infinite number of energy levels: the quantum Harmonic Oscillator ( ). The quantum is a model that describes systems with a characteristic energy spectrum, given by a ladder of evenly spaced energy levels. The energy difference between two consecutive levels is E.

31 For more details on Hermite Polynomials and their generator function, look on Cohen-Tannoudji. Online information from: Eric W. Weisstein. Hermite Polynomial. From MathWorld–A Wolfram Web Resource. 82. 9.1.4 Heisenberg picture We want now to study the time-evolution of the h.o. We first start with analyzing the evolution of the operators in

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