Tight binding method for a 1D crystal with a diatomic basis

[adamoob]

Hi, I'd be most grateful for any help regarding the following problem:

Consider a 1D crystal with 2 atoms in a primitive cell (let's call them atoms A and B). Each atom has only one valence orbital denoted as $\left|\psi_A(n)\right>$ and $\left|\psi_B(n)\right>$ respectively.

Show that the superposition
$\left|k\right> = \sum_{n = -\infty}^{\infty}e^{ikna}\left[ c_A\left|\psi_A(n)\right> + c_B\left|\psi_B(n)\right>\right]$
satisfies the Bloch's theorem ($c_A, c_B$ are constant)
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I know that the theorem is usually formulated as follows:
There is a $\vec{k}$ from the reciprocal lattice for each wavefunction so that
$\psi\left( \vec{r}+\vec{R}\right) = e^{i\vec{k}\vec{R}}\psi\left( \vec{r}\right)$
$\vec{R}$ is a vector from the direct lattice. But I still don't know what to do.

I am probably not very good with quantum mechanics yet (we had the solid state and QM simultaneously this semester) and I realize this may be very easy. But I have been stuck on this and will be grateful for any help.

 

[eljose79]

Hi there,

I am happy to help you with this problem. First, let's start by understanding Bloch's theorem. It states that the wavefunction of an electron in a periodic potential can be written as a product of a plane wave and a periodic function. In other words, the wavefunction has the form of e^{i\vec{k}\vec{r}}u(\vec{r}), where e^{i\vec{k}\vec{r}} is the plane wave and u(\vec{r}) is the periodic function.

Now, let's apply this to your problem. We have a 1D crystal with 2 atoms in a primitive cell, and each atom has one valence orbital. This means that the potential is periodic with a period of a, the lattice constant. We can write the wavefunction in the form of e^{i\vec{k}\vec{r}}u(\vec{r}), where e^{i\vec{k}\vec{r}} is the plane wave and u(\vec{r}) is the periodic function.

Next, we can use the superposition given in the problem to write the wavefunction as a linear combination of the basis states \left|\psi_A(n)\right> and \left|\psi_B(n)\right>. This is exactly what is given in the problem, where c_A and c_B are the coefficients of the linear combination. Now, we can rewrite this as:

\left|k\right> = \sum_{n = -\infty}^{\infty}e^{ikna}\left[ c_A\left|\psi_A(n)\right> + c_B\left|\psi_B(n)\right>\right] = \sum_{n = -\infty}^{\infty}e^{i\vec{k}\vec{r}}\left[ c_A\left|\psi_A(n)\right> + c_B\left|\psi_B(n)\right>\right]

We can see that the first term in the sum is the plane wave e^{i\vec{k}\vec{r}}, and the second term is the periodic function u(\vec{r}). This satisfies Bloch's theorem, with \vec{k} being the reciprocal lattice vector and u(\vec{r}) being the periodic function.

I hope this helps you understand how the given superposition satisfies Bloch's theorem. Let me know if you have any further questions. Good luck