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The potential inside the crystal is periodic ##U(\vec{r} + \vec{R}) = U(\vec{r})## for lattice vectors ##\vec{R} = n_i \vec{a}_i##, ##n_i \in \mathbb{Z}## (where the ##\vec{a}_i## are the crystal basis), and Hamiltonian for an electron in the crystal is ##\hat{H} = \left( -\frac{\hbar^2}{2m} \nabla^2 + U(\vec{r}) \right)##. The book defined a translation operator ##T_{\vec{R}}##, and proved that ##T_{\vec{R}}## and ##\hat{H}## are commuting operators, so have simultaneous eigenstates,$$\hat{H} \psi = E\psi, \quad T_{\vec{R}} \psi = c(\vec{R}) \psi$$They also prove that ##c(\vec{R} + \vec{R}') = c(\vec{R})c(\vec{R}')## for two lattice vectors ##\vec{R}## and ##\vec{R}'##. But I don't understand the next bit, which says ##c(\vec{a}_i)## can always be written in the form$$c(\vec{a}_i) = e^{2\pi i x_i}$$for some ##x_i##. I feel like I'm missing something obvious, why can we do this? Thanks!

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