What happens at energy-gaps, Kronig-Penney model

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SUMMARY

The discussion focuses on the behavior of wave-functions at energy gaps as described by the Kronig-Penney model in one-dimensional periodic boundaries. It is established that wave-functions do not need to be continuous as a function of energy (E), with atomic and molecular spectra providing counterexamples. In the presence of a periodic potential, the degeneracy of wave-functions associated with free electrons is lifted, resulting in eigenfunctions such as cos(kx) and sin(kx) corresponding to different energy levels.

PREREQUISITES
  • Understanding of the Kronig-Penney model
  • Familiarity with wave-functions and their properties
  • Knowledge of periodic potentials in quantum mechanics
  • Basic concepts of energy bands and band gaps
NEXT STEPS
  • Study the implications of periodic potentials on wave-functions in quantum mechanics
  • Explore the concept of degeneracy lifting in quantum systems
  • Learn about the relationship between wave-vector (k) and energy (E) in solid-state physics
  • Investigate the role of eigenfunctions in determining energy levels in quantum systems
USEFUL FOR

Students and researchers in quantum mechanics, particularly those focusing on solid-state physics, as well as anyone interested in the implications of the Kronig-Penney model on wave-function behavior and energy gaps.

iAlexN
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I'm studying periodic boundaries in 1D for which energy gaps occur, i.e. values of the energy which are not allowed, according to the Kronig-Penney model. My question is when these band gaps occur what happens to the wave-function, because it still has to be continuous, so it can't just terminate, so does it stay the same? Or what happens to the wave-function at these regions?

Thanks!
 
The wavefunctions don't have to be continuous as a function of E, the spectra of atoms or molecules serve as a counter example. For free electrons, the wavefunctions are somehow continuous as a function of k and E, but, there are always two degenerate solutions with the same value of E, namely those with k and -k. The periodic potential will lift this degeneracy, i.e. while for a free particle the functions exp ikx and exp -ikx, or equivalently, cos kx and sin kx are degenerate, with a weak potential, the functions cos kx and sin kx are eigenfunctions belonging to different energies.
 

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