Band Spectrum in a Periodic Potential

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Plot of the symmetric solution of .

Contributed by: Michael Trott (January 2011)
Open content licensed under CC BY-NC-SA



is the energy.

The spectrum of the Schrödinger operator with a periodic potential has allowed energy bands. When the energy is within a band, the eigenfunctions are the product of a phase factor of magnitude 1 and periodic parts. As a result, the eigenfunctions are ‐normalizable. When the energy is outside the bands, the eigenfunctions grow unbounded as . You can see both situations by varying the energy.

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