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.