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Consider Kroenig Penney model. Here there is a derivation that uses the following potential:

enter image description here

One of the assumption is that the energy of the electron is $E<U_0$. Under this assumption (and others) the following relation is derived:

$$\displaystyle \cos(ka)=\cos(\alpha a)-P{\sin(\alpha a) \over \alpha a}\qquad \left(P={\beta ^{2}ab \over 2}\right)\,\!$$

Which basically explains the band structure of energies of electron in a material.

My question is : what is different in the case where $E>U_0$? In that case is the band theory still valid?

On Alonso Finn textbook the following picture is discussed (it's a periodic potential). It is said that the electron can have different energies, for example E1 E2 and E3. The problem is with E3: I can't understand if it is the case of a metal with electron in conduction band or if it is a different case.

So is the band theory valid for electron with energy E3? Is it simply an electron in conduction band (and therefore following the band theory), or not?

enter image description here

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Band theory is valid when $E>U_{0}$. You can repeat the derivation for the case where $E<U_{0}$, but now the wavefunction will be plane wave everywhere instead of being decaying in the barrier.

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