Consider an observable with a spectrum that has both a discrete and continuous part. If a measurement is made for this observable while the system is in some state, if the measurement outcome was one of the eigenvalues of the observable, the state immediately after the measurement is the one generated by any complex multiple of the eigenvector corresponding to the eigenvalue that was observed. If the measurement outcome came to lie in the continuous part of the spectrum, what would the state after the measurement be? Would it be a mixed state of some sort? What exactly happens in this situation?
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Possible duplicate: Measurement of observables with continuous spectrum: State of the system afterwards – Chiral Anomaly Apr 29 '21 at 23:54
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For example, consider a particle in a finite square potential well, with well depth V. Then the energy spectrum can be continuous (for free particle state when E>0) and discrete (for particle with -V<E<0, corresponding to an eigenstate of the square well). Now, if the measurement outcome of energy lies in the continuous part of the energy spectrum, the particle state would collapse to the free particle eigenstate with energy eigenvalue E=$\hbar^2k^2/2m$.
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