513.001 Molecular and Solid State Physics

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Physical
Constants

Periodic System
of Elements

      

Bloch wave

For a certain periodic one-dimensional potential, a solution to the Schrödinger equation is known to be,

\[ \begin{equation} \psi_k(x)=e^{ikx}\cos\left(\frac{2\pi x}{a}\right). \end{equation} \]

In this case, $\frac{\pi}{a} < k < \frac{2\pi}{a}$ is outside the first Brillouin zone. This solution can be written in the form, \[ \begin{equation} \psi_{k'}(x)=e^{ik'x}u_{k'}(x) \end{equation} \]

Where $u_{k'}(x)$ is a periodic function and $k'$ is in the first Brillouin zone. What is $k'$ and what is $u_{k'}(x)$?

Hint: Write cosine as the sum of two complex exponentials.