Pre-lecture problem (Homework 1c).

Here is a problem that you can do if you like. It is optional, but if you want to email me your answer to the last part (what is it as a fraction of the speed of light, c?) you'll get extra credit. It provides a chance to calculate actual numbers again and it is related to something we will cover tomorrow in class.

5. Consider the expression $E_{1,k} = E_1 -2 \gamma_1 cos(ak)$. Suppose gamma = 2 eV.
Calculate the group velocity  $v(k) = \frac{1}{\hbar}\frac{\partial E_{1,k} }{\partial k }$. Graph it vs k and evaluate it at its maximum value. I think it is at $k = \pi/2a$. What is it as a fraction of the speed of light, c?
hint:

multiply top and bottom by c and use $\hbar c =197$ eV-nm like before (see HW 1a post)