Consider a one-dimensional crystal of atoms with one valence orbital $\phi$. The tight binding wavefunction for this case is,
\[ \begin{equation} \psi_{k}\left(x\right)=\frac{1}{\sqrt{N}}\sum\limits_{n}e^{inka} \phi\left(x-na\right). \end{equation} \]Here $n$ is an integer. Multiply the Schrödinger equation from the left by $\phi^*\left(x\right)$ and integrate over all space.
\[ \begin{equation} \langle\phi\left(x\right)|\hat{H}|\psi_{k}\left(x\right)\rangle = E\langle\phi\left(x\right)|\psi_{k}\left(x\right)\rangle. \end{equation} \]In the tight binding approximation, only the on-site and nearest neighbor matrix elements are retained on the left side and only the on-site term is retained on the right side. It is assumed that all the other terms are small enough that they can be ignored.
\[ \begin{equation} \langle\phi\left(x\right)|\hat{H}|\phi\left(x-a\right)\rangle e^{-ika} +\langle\phi\left(x\right)|\hat{H}|\phi\left(x\right)\rangle+\langle\phi\left(x\right)|\hat{H}|\phi\left(x+a\right)\rangle e^{ika} + \text{small terms}= E + \text{small terms} . \end{equation} \]Let $\epsilon = \langle\phi\left(x\right)|\hat{H}|\phi\left(x\right)\rangle$ and $t = - \langle\phi\left(x\right)|\hat{H}|\phi\left(x-a\right)\rangle$. The dispersion relation can then be written as,
\[ \begin{equation} E= \epsilon -t\left(e^{-ika} + e^{-ika}\right). \end{equation} \]The exponential terms can be combined into a cosine term.
\[ \begin{equation} E= \epsilon -2t\cos\left(ka\right) . \end{equation} \]The form below generates a table of where the first column is the wavenumber $k$ in 1/m and the second column is the energy $E$ in units of eV.
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