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===== Prerequisites =====
Before beginning this topic, students should:


  * Know how to represent the displacement of a chain of monatomic molecules using an envelope function.
  * Have some experience in quantum mechanics, including using the energy eigenvalue equation and Schrodinger's equation.
  * Understand how to represent quantum states using bra-ket notation or function notation.
  * Be able to comfortably transition between bra-ket notation and function notation.
  * Have experience with the properties of observables.
  * Know how to find the eigenvectors and eigenvalues for a matrix.

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