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Phonon's dispersion relation can be obtained directly using the quantum-mechanical approach. The problem here is to find the solution of the Schrodinger equation for the lattice vibrations. The quantum-mechanical operators for the normal mode coordinates are substituted in the phonon Hamiltonian by the two operators, called creation and annihilation operators. The names of these operators arise because their effects are to create and annihilate phonons respectively.
The form of the Hamiltonian operator and the wave function are not shown here, however, some conclusions of the quantum-mechanical approach are listed below:.
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