Bloch function - Solid state physics

In summary, Bloch functions are one-electron wavefunctions that can be used in the quantum mechanical modelling of crystals. They can be assembled into wave packets to represent freely propagating electrons through the potential field of the ion cores. By using a plane wave basis set and calculating at different crystal moments, we can determine the electronic band structure and obtain various properties of the solid.
  • #1
big man
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Homework Statement


What are Bloch functions? Explain briefly how these function are used in the quantum mechanical modelling of crystals.

Homework Equations


[tex]\psi _k (r)=u_k(r)exp(ik.r)[/tex]


The Attempt at a Solution


Now I know that a bloch function is a one-electron wavefunction that is given by the above equation. But I don't really know how to explain how it is used for the quantum mechanical modelling of crystals.

Kittel gives this information on a bloch function:
Bloch functions can be assembled into wave packets to represent electrons that propagate freely trhough the potential field of the ion cores.

To me this description answers it and hell the equation gives the relationship between the electron wavefunction with the periodicity of the crystal lattice so I would have thought that, that is quantum mechanic modelling in itself. Is there something else I should be putting down or have I just interpreted this incorrectly.

Any help is greatly appreciated.
 
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  • #2
##u_r## and ##e^{ikr}## are both periodic by Bloch’s theorem, so we can use a plane wave basis set to approximate the actual wavefunction of the system. By calculating at a number of crystal moments, we can work out the electronic band structure of the material, which we can use to get a long list of properties of the solid.
 

1. What is a Bloch function in solid state physics?

A Bloch function is a mathematical representation of an electron's wave function in a crystalline solid. It describes the probability of finding an electron at a particular point in the crystal lattice, taking into account the periodic nature of the lattice.

2. How is a Bloch function different from a free electron wave function?

A Bloch function takes into account the periodic nature of the crystal lattice, while a free electron wave function does not. This means that a Bloch function is only valid within a specific crystal lattice, while a free electron wave function can describe an electron in any location.

3. What is the significance of Bloch functions in solid state physics?

Bloch functions are important because they provide a mathematical description of how electrons behave in a crystalline solid. This allows us to understand the electronic properties of materials and how they relate to the structure of the crystal lattice.

4. How are Bloch functions related to band structure in solids?

The band structure of a solid is determined by the energy levels of the electrons in the crystal lattice. Bloch functions are used to calculate these energy levels, as they describe the allowed energy states for electrons within the lattice.

5. Can Bloch functions be used to study the electronic properties of all types of solids?

No, Bloch functions are only applicable to crystalline solids. Amorphous materials, such as glasses, do not have a periodic crystal lattice and therefore cannot be described using Bloch functions.

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