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In solid-state physics, we often encounter the velocity operator (e.g. this post) which the eigenvalues can be written as $$ V^\mu(\vec{k}) = \frac{1}{\hbar} \frac{\partial E}{\partial k^\mu} \biggr|_{\vec{k}} $$

Is there a physical interpretation for the second derivative of $E$ with the components of $\vec{k}$? $$ \frac{\partial^2 E}{\partial k^\mu \partial k^\nu} $$

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    $\begingroup$ The effective mass. $\endgroup$
    – Tobias Fünke
    Commented Jun 1 at 18:40
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    $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$
    – hft
    Commented Jun 1 at 18:50
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    $\begingroup$ Why would someone vote for "needs details or clarity"? The question is crystal (no pun intended) clear. $\endgroup$
    – Tobias Fünke
    Commented Jun 1 at 19:31
  • $\begingroup$ Thanks, @TobiasFünke can you turn that comment into a short answer? $\endgroup$
    – Bio
    Commented Jun 4 at 5:47
  • $\begingroup$ No problem. I don't have much time. Consider to write an answer yourself, perhaps with a bit of explanation! In fact, it is very welcomed to do so. You can also accept your own answer. $\endgroup$
    – Tobias Fünke
    Commented Jun 4 at 5:57

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