Solution (source code)

= Solution

Let the missing electron occupy the <Bloch state> of <crystal momentum> $\hbar\mathbf k_e$ and <energy> $E_e(\mathbf k_e)$. Relative to the filled band, define $\mathbf k_h=-\mathbf k_e$ modulo a reciprocal-lattice vector and $E_h(\mathbf k_h)=C-E_e(-\mathbf k_h)$, with a fixed reference $C$. Then
$$
\boxed{\mathbf p_h=-\mathbf p_e,\quad E_h=C-E_e,\quad\mathbf v_h=\mathbf v_e,\quad(m_h^{*-1})_{ij}=-(m_e^{*-1})_{ij},\quad q_h=-q_e.}
$$
The velocity equality follows by differentiating the hole energy with respect to $\mathbf k_h$; two minus signs cancel. The signed <band effective mass> is therefore opposite, becoming a positive hole mass near an electron-band maximum. The hole <electric charge> is $+e$ when the electron charge is $-e$. These momenta are crystal momenta, and the electron mass being compared is the signed band mass, not the bare electron mass. The positive hole current equals the current lost by removing that electron.