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=====Differences Between Vibrating Atoms and Electrons in a Crystal (5 minutes)=====
  * Vibrations of atoms in a crystal:
    - Involves analyzing discrete periodic system with many springs and masses.  The movement of the masses is described using envelope functions.                                                                                                                {{courses:lecture:pplec:pplecdifferencesfig1.png|(Add image: pplecdifferences1)}}
    - Boundary conditions result in normal modes (envelope functions where all atoms move with the same frequency).
    - Time dependence: $e^{i \omega t}$.

  * Electrons in a crystal:
    - Involves analyzing a periodic system with one electron and many atoms.  The probability of finding one electron somewhere amongst the atoms is described using envelope functions.                                                                       {{courses:lecture:pplec:pplecdifferencesfig2.png|(Add image:pplecdifferences2)}}
    - Boundary conditions result in eigenstates (eigenstates are the analog of normal modes).
    - Time dependence evolves from the Schrodinger Equation: $e^{i\frac{E}{\hbar}t}$.

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