Suppose $\psi_{nk}=e^{ikr}u_{nk}$ is the Bloch function of a periodic Hamiltonian $H(r)=H(r+R)$, where $H(r)\psi_{nk}=\varepsilon_{nk}\psi_{nk}$ and $H(k)=e^{-ikr}H(r)e^{ikr}$.

What would the following quantity change under the time-reversal and inversion operation respectively and why?

$$ \vec{L}(k)=i\int \mathrm{d}r \nabla_k u_{nk}^*\times\left[H(k)-\varepsilon_{nk}\right]\nabla_k u_{nk} $$

I have no clue how to tackle this, I don't know if I am going to change $u_{nk}$ or to change $H(k)$ and $\varepsilon_{nk}$ or change both, and I am not sure how to change them. Please give a detailed answer if you know how to do this.


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