Transcription of 9. Harmonic Oscillator - MIT OpenCourseWare
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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.
9.1.1 Classical harmonic oscillator and h.o. model A classical h.o. is described by a potential energy V = 1kx2. If the system has a finite energy E, the motion is bound 2 by two values ±x0, such that V(x0) = E. The equation of motion is given by mdx2 dx2 = −kxand the kinetic energy is of course T= 1mx˙2 = p 2 2 2m. The energy is constant ...
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