Simplest tight binding

Question

My lecturer is teaching the Bloch theorem, which I saw many years ago in Griffiths textbook on Quantum mechanics. I cannot recognise it in the form that my lecturer is using.

We are studying a tight-binding model for a one-dimensional chain of atoms. This is what my lecturerer writes:

Consider a one-dimensional chain of (sodium) atoms, each one represented by a fix point charge of +1. In the search for the eigenstates for an electron in interaction with these charges we resort to a basis of one s-like orbital per atom, that we will call $X$ , indicating where it is located, $X = j a$, with $a$ the lattice parameter and $j$ an integer indicating the lattice site. Let us make an additional approximation: The Hamiltonian matrix elements will be considered non-zero only when involving neighbouring sites, the nearest-neighbour approximation, i.e., $\langle X | H | X \rangle = \varepsilon$ and $\langle X +a | H | X \rangle = -t$. Therefore the standard matrix of the hamiltonian in this basis is tridiagonal: $$ H = \begin{bmatrix} \ddots& \\ &\varepsilon & -t & 0 \\ &-t& \varepsilon & -t & \\ & 0 &-t&\varepsilon&\\ &&&& \ddots \end{bmatrix} $$ The solution is trivial if we construct Bloch states as linear combinations of the basis states: $$ |k\rangle = N^{-1/2} \sum_{X}^N e^{ikX} |X \rangle $$ For the basis set we have, with one single basis state per unit cell, there is only one such Bloch state per k value. Our Hamiltonian will thus be a block diagonal matrix with blocks 1x1: It will be directly diagonal!

I have some questions about this:

• How can my lecturer assume that if $|k \rangle$ is a bloch function, that there exists a formula like this: $|k\rangle = N^{-1/2} \sum_{X}^N e^{ikX} |X \rangle$? The bloch theorem says that there exists a fuction $u$ that can take the role of $|X \rangle$ in this sum, if I (falsely) assume that $|X\rangle$ is a basis. Does he arrive at this in some other way?
• My lecturer says that there is only one Bloch state per k value. Is this to say that there is exactly one band? If so, how can we prove this?

Answer 1

Well, I've come to study this from a chemistry perspective recently. So here is my take on the first two questions.

For the first question: $|k\rangle$ is the Bloch function formed from atomic orbital localized on each atoms. Bloch Theorem is based on the translational invariance of the system, so that's why we just use a single $|X\rangle$ but translate it with the exponential factor.

For the second question: Notice that if we fix $k$, then the basis set simply consists of the Bloch functions formed from individual atomic orbitals. So the number of basis state, or Bloch state, whatever you call it, is just equal to the number of atomic orbitals considered in the problem. Since your instructor is just considering one s-like orbital, you only have one basis state and one band.