## Differences Between Vibrating Atoms and Electrons in a Crystal (5 minutes)

• Vibrations of atoms in a crystal:
1. Involves analyzing discrete periodic system with many springs and masses. The movement of the masses is described using envelope functions.
2. Boundary conditions result in normal modes (envelope functions where all atoms move with the same frequency).
3. Time dependence: $e^{i \omega t}$.
• Electrons in a crystal:
1. 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.
2. Boundary conditions result in eigenstates (eigenstates are the analog of normal modes).
3. Time dependence evolves from the Schrodinger Equation: $e^{i\frac{E}{\hbar}t}$.

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