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A random question struck my mind. If $|\phi_n\rangle$ be an eigenstate of Hamiltonian H (s.t. $H |\phi_n\rangle =E_n|\phi_n\rangle$), so is $e^{i\delta_n}|\phi_n\rangle$ for any real $\delta_n$. This is true for any n. These phases aren't global but relative, as they depend on $n$. Would these ambiguous phases that could be attached to eigenstates be problematic while making predictions using these states?

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No. Such arbitrary phase choices make no difference. Physical quantities are derived from probabilities that are computed from the magnitude of squares of matrix elements $$ |\langle n|O|m\rangle|^2 $$ and these are unaffected by the phases. Phase choices do get involved in the formalism sometimes. For example in the adiabatic theorem the "Berry phase" depends on the phase choices. But again, when one computes a physical quantity, the phases drop out in the end.

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  • $\begingroup$ Should it not be the Berry connection which depends on the phase choices, with the Berry phase being left invariant under such changes? $\endgroup$
    – J. Murray
    Aug 15, 2020 at 16:54
  • $\begingroup$ @J. Murray. Yes. You are correct! $\endgroup$
    – mike stone
    Aug 15, 2020 at 18:31

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