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1. The problem statement, all variables and given/known data

In Figure 3 of Chapter 7 of Kittel's Solid-state physics book, it says that this is the key to understanding the origin of the energy gap. However I do not understand why. It seems like if you take the expectation value of either [tex]|\psi(-)|^2[/tex] or [tex]|\psi(+)|^2[/tex], you will get exactly the same value. How can shifting the phase of the wave change its expectation value? There is absolutely no reason why the expectation value of sine squared should be different than the expectation value of cosine squared!

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Kittel page 166

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