What happens at energy-gaps, Kronig-Penney model

[iAlexN]

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!

 

[DrDu]

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.