# Bloch Functions: Explaining the Bloch-Floquet Theorem

• danja347
In summary, the theorem states that solutions to periodic variations in the refractive index must themselves be periodic, and that this phase can only differ by a phase depending on the lattice vector.
danja347
For propagation in a periodic dielectric crystal i can by combining Maxwells equations under certain conditions get:

$$\bold{\nabla}\times{1\over\epsilon(\bold{x})}\bold{\nabla}\times\bold{H}=\left({\omega\over{c}}\right)^2\bold{H}$$

I can apply Bloch-Floquet theorem and then draw a lot of conclusion.
Where does Bloch-Floquet theorem come from, when can I apply it and how can it be explained?

Thanks!

/Daniel

Why not try google.For example the first search resualt I got is:

http://www.elettra.trieste.it/experiments/beamlines/lilit/htdocs/people/luca/tesihtml/node7.html

Last edited by a moderator:
The essence of the theorem is, that any solution to a periodic variation in the refractive index, must itself be periodic (To me, this is a fairly logical conclusion).

There is a wealth of information about this theorem, as it is not only used in photonic crystals, but also to study bandgaps in metals and semi-conductors.

As for the google, I thought the 3rd one down was pretty good (A little more layman for those that don't have a large mathematics background).

Claude.

Also, if you have access to a library, find 'Solid State Physics' by Ashcroft & Mermin or Kittel. Both have derivations of Bloch's Theorem.

Here's an insight into Bloch's theorem that most texts do not mention:

The idea is that in a period potential, the probablilty of finding an electron at some location should be equal to the probablity of finding the electron at all other places which are identical due to periodicity- and this makes sense. Here's the punchline - since the lwavefunctionl^2 gives the probability, this means that at all those places the wave function can differ only by a phase. The not so obvious thing is that this is not just any phase, but a phase whose argument is a function of the lattice vector.

Thank you all for you replies... I think I am getting a better and better understandning about how things work!

So, thanks again!

/Daniel

## 1. What is the Bloch-Floquet Theorem?

The Bloch-Floquet Theorem is a mathematical concept that explains the periodicity and symmetry of Bloch functions in solid state physics. It states that a wave function in a periodic potential can be written as a product of a periodic function and a plane wave.

## 2. How does the Bloch-Floquet Theorem relate to solid state physics?

The Bloch-Floquet Theorem plays a crucial role in understanding the electronic properties of crystalline solids. It explains how the electrons in a crystal behave in a periodic potential, leading to the formation of energy bands and the electrical conductivity of the material.

## 3. What are Bloch functions?

Bloch functions are solutions to the Schrödinger equation for a particle in a periodic potential. They have a periodicity that matches the periodic lattice of the crystal, and are characterized by a wave vector and a Bloch phase factor.

## 4. What is the significance of the wave vector in Bloch functions?

The wave vector, also known as the Bloch vector, represents the momentum of an electron in a crystal. It determines the energy of the electron and its position in the Brillouin zone, which is a representation of the periodic lattice in reciprocal space.

## 5. How does the Bloch-Floquet Theorem impact our understanding of materials?

The Bloch-Floquet Theorem is essential for understanding the electronic and optical properties of materials. It allows us to predict and explain the behavior of electrons in a crystal, which is crucial for developing new technologies and materials with specific properties.

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