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When using plane wave basis functions,

what is the phase of each plane wave?

Should a one-electron wavefunction looks like $\Phi^k_{\beta}(r) = \sum_{G} c^k_{\beta,G}e^{i((k+G) r + \varphi_G)}$ instead of $\Phi^k_{\beta}(r) = \sum_{G} c^k_{\beta,G}e^{i(k+G) r}$?

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It does not matter: the coeficients are supposed to be complex, absorbing the phases.

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    $\begingroup$ Even more, the phase does not affect the plane wave: exp(iφ)exp(ikr) is physically indistinguishable from exp(ikr) itself. $\endgroup$
    – elutionary
    Commented Nov 21, 2023 at 13:51
  • $\begingroup$ When solving the eigenvalue problem of the Hamiltonian, are the coefficients complex or real? (The 'could' in the answer is disturbing me.) $\endgroup$
    – mle
    Commented Nov 21, 2023 at 18:56
  • $\begingroup$ It depends on what problems you are solving @mle, you could met cases where all coefficients are real. But in general (in symbolic calculation), you should treate them as complex, i.e., using their adjoint, not transpose when switch from $\psi$ to $\psi^\ast$. $\endgroup$
    – Ryan
    Commented Nov 24, 2023 at 3:04

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