Understanding the Origin of Energy Gap in Kittel's Book

[ehrenfest]

[SOLVED] kittel page 166

Homework Statement

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 $$|\psi(-)|^2$$ or $$|\psi(+)|^2$$, 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!

Homework Equations

The Attempt at a Solution

 

[malawi_glenn]

I can not see where he his saying that the expacation values of the two wave functions in space are equal. He states: "When we caclulate the expectation values of the POTENTIAL ENERGY over the the ..:"

Look at the potential!... and do a quick head calculation.

psi(+) has its peaks right "above" the ion cores, i.e where the potential energy is max (see fig a). So psi(+) has larger probability to be located where the pot E is small ( = minus infinity). And psi(-) have its peakes "between" the ion cores, where the pot E is max ( = 0).

So the average energy of the the psi(+) is shifted down in comparison with psi(-), and that gives you the band gap.

 

[ehrenfest]

That makes sense! You are calculating $$\int |\psi(x)|^2 V(x) dx$$ not $$\int |\psi(x)|^2 dx$$ since the latter is just 1.

 

[malawi_glenn]

exactly :)