Why Isn't Bloch's Theorem Reduced to Unity?

In summary, the Bloch theorem states that the wave function, psi, at a position r plus a lattice vector R is equal to the phase factor e^ikR times psi at position r. The k vector is in the reciprocal lattice and satisfies the condition e^KR = 1. This means that k is not limited to a reciprocal lattice vector and only applies to standing waves. The complex phase of the wave gives it a direction.
  • #1
toqp
10
0
This is not any homework problem but just something I don't understand. The Bloch theorem states that
[tex]\psi(\textbf{r}+\textbf{R})=e^{i\textbf{k}\cdot \textbf{R}}\psi(\textbf{r})[/tex]

Now the k is a vector in the reciprocal lattice (usually in the first Brillouin zone), which is defined as the set of vectors K that satisfy
[tex]e^{i\textbf{K}\cdot\textbf{R}}=1[/tex]

Now, if k points to a point in the reciprocal lattice, then why isn't the Bloch theorem
[tex]\psi(\textbf{r}+\textbf{R})=e^{i\textbf{k}\cdot \textbf{R}}\psi(\textbf{r})[/tex]
just
[tex]\psi(\textbf{r}+\textbf{R})=1\psi(\textbf{r})[/tex]?
 
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  • #2
AFAIK, k is not limited to a reciprocal lattice vector. I think that would only be for standing waves. The complex phase gives the wave a direction.
 

FAQ: Why Isn't Bloch's Theorem Reduced to Unity?

What is the Bloch Theorem?

The Bloch Theorem is a fundamental principle in solid state physics that explains the behavior of electrons in a periodic crystal lattice.

Why is the Bloch Theorem important?

The Bloch Theorem provides a framework for understanding the electronic structure of materials, which is crucial for a wide range of applications in science and technology, such as designing new materials for electronics and nanotechnology.

What does the Bloch Theorem state?

The Bloch Theorem states that the wavefunction of an electron in a periodic crystal lattice can be written as the product of a plane wave and a periodic function, known as the Bloch function.

What are the assumptions of the Bloch Theorem?

The Bloch Theorem assumes that the crystal lattice is infinite, the potential is periodic, and the electrons are non-interacting.

How does the Bloch Theorem explain the electronic properties of materials?

The Bloch Theorem explains how the periodic potential of a crystal lattice affects the energy levels and motion of electrons, leading to properties such as electrical conductivity and magnetism.

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