# Understanding the Origin of Energy Gap in Kittel's Book

• ehrenfest
In summary, the conversation discusses the key to understanding the origin of the energy gap in Figure 3 of Chapter 7 of Kittel's Solid-state physics book. The speaker is confused about how shifting the phase of a wave can change its expectation value, but it is explained that the potential energy plays a role and that the average energy of one wave is shifted down compared to the other, resulting in the energy gap. The expectation values of the two wave functions in space are not equal and this is important in calculating the potential energy.

#### 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!

## The Attempt at a Solution

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 probabilty 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.

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.

exactly :)

## 1. What is the significance of understanding the origin of energy gap in Kittel's book?

The energy gap in Kittel's book refers to the energy difference between the highest occupied and lowest unoccupied energy states in a solid material. Understanding the origin of this energy gap is crucial in understanding the electronic and optical properties of materials, which have a direct impact on their use in various technological applications.

## 2. What is the origin of energy gap in Kittel's book?

The energy gap in Kittel's book is a result of the quantum mechanical behavior of electrons in a solid material. It arises from the discrete energy levels that electrons can occupy in a crystal lattice, as well as the interactions between these electrons and the atoms in the lattice.

## 3. How does the energy gap in Kittel's book relate to band structure?

The energy gap in Kittel's book is closely related to the concept of band structure, which describes the distribution of allowed energy levels for electrons in a solid material. The energy gap determines the separation between the valence and conduction bands, which are responsible for the electrical and optical properties of a material.

## 4. What factors influence the size of the energy gap in Kittel's book?

The size of the energy gap in Kittel's book is determined by several factors, including the type of material, the crystal structure, and the presence of impurities or defects. Additionally, external factors such as temperature and pressure can also affect the size of the energy gap.

## 5. How is the energy gap in Kittel's book experimentally determined?

The energy gap in Kittel's book can be experimentally determined using various techniques, such as spectroscopy, electrical conductivity measurements, and photoemission spectroscopy. These methods involve measuring the energy required to promote an electron from the valence band to the conduction band, which corresponds to the size of the energy gap.