 Quote by silverwhale
alxm, do you mean by same observables the same expectation values of the observables?
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Expectation values, then.
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If I understood you right the phase IS the eigenvalue of P. And we obtain this equating the first equation derived from the expectation values to the eigenvalue equation of P.
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Switching twice must result in the same function, so the phase factor squared must equal unity, hence -1 or 1.