Lecture 21: Periodic Lattices Part 2

TL;DR

The energy eigenvalues and eigenfunctions of a particle in a periodic potential form bands and have gaps between them, due to interference effects caused by scattering off the lattice.

Transcript

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Key Insights

• 🤕 The energy eigenvalues and eigenfunctions in a periodic potential form bands separated by gaps.
• 👋 Wave packets must be used to study localized states in a periodic potential.

Questions & Answers

Q: What happens to the energy bands and gaps as the periodic potential becomes stronger?

As the potential becomes stronger, the energy bands get narrower and the gaps get wider.

Q: Why do we have bands and gaps in the energy eigenvalues of a periodic potential?

The bands and gaps arise due to interference effects caused by scattering off the lattice, which determines the allowed energy eigenvalues.

Q: What happens to the velocity of a wave packet as the crystal momentum q increases?

In the vicinity of the bottom of a band, the velocity increases as q increases. However, at higher energies, the velocity decreases as q increases.

Summary & Key Takeaways

• The energy eigenvalues are restricted to lie within bands separated by gaps in a periodic potential.

• The energy eigenfunctions are extended and wave packets must be used to study localized states.

• The group velocity of wave packets can be calculated using the dispersion relation, and the effective mass can be computed from the momentum and group velocity.