Suppose that we have a nondegenerate quantum system with an orthonormal basis of eigenstates $|\psi_n\rangle$ with eigenvalues $E_n$. Consider a wavefunction $\psi(x,t)=\sum_n c_n|\psi_n(x,t)\rangle$. Suppose that we measure the energy at time $T$ and obtain value $E_m$. Now the system collapses to the eigenstate $|\psi_m\rangle$. However, what will be the new phase factor? The new wavefunction must be equal to $$A|\psi_m\rangle$$ for some complex number $A$ with $|A|=1$. Is there a rule to determine the value of $A$? With a `rule', I mean a mathematical formula with the numbers $c_n$ and the physical properties of the system as input and $A$ as output.
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If the measurement measures phase, e.g., using a kind of Aharonov-Bohm device, then the phase constitutes a quantum number in the spectrum of the measurement operator - that is, in the process of measurement we project the wave function on a state with a specific phase (usually known up to an additive constant).
Otherwise the phase is not relevant (not determined), as it will not influence the results of measurement.
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$\begingroup$ Maybe the phase is relevant for future measurements: for example, if there will be another particle which will interfere with the particle in the future, then the phase determines how the particles will interact (amplify or cancel out). $\endgroup$– RiemannJan 27 at 14:51
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$\begingroup$ I am interested in the case where the measurement is of energy, not of phase. $\endgroup$– RiemannJan 27 at 14:54
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$\begingroup$ @Riemann For subsequent measurements your current measurement is just a a physical process that somehow changes the wave function. If you can fully describe this process (specifying the measurement Hamiltonian, interactions with environment, etc.) then determining the phase for future measurements is a matter of solving the equations. If however you suggest that some information is lost in your measurement, then all the bets are off. $\endgroup$– Roger V.Jan 27 at 15:16
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$\begingroup$ So you are saying that if you know the process of measurement, then you can calculate $A$ using some rule? $\endgroup$– RiemannJan 27 at 16:58
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$\begingroup$ @Riemann but what is measurement in your case? It is also a physical process - it somehow changes the phase, but you have no information about it, since you do not measure it. $\endgroup$– Roger V.Jan 27 at 17:04