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Is it possible that the incoming particles, in a scattering experiment, have a superposition state of up and down spin? In this case, aren't we supposed, after we calculate the matrix elements for the different spin states, to add these amplitudes rather than their squares?

In other words, why do we always do this $\sum_{ss'}\left|\mathcal{M}\right|^2$ rather than $\left|\sum_{ss'}\mathcal{M}\right|^2$?

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There are two types of uncertainty. The first is the classical kind where the system is in a particular state and we don't know what that state is. In that case one should add the probabilities which are given by the magnitude of the amplitude squared. The second type of uncertainty is the quantum mechanical one. In this case the system is not in one particular state but instead has different amplitudes for more than one state. Then, if one want to work out the probability, one first adds over the amplitudes and then takes the magnitude squared of the result. The case you discuss is of the first kind as the incoming particles are in a particular spin state, we just don't know what that spin state is.

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  • $\begingroup$ , following the answer by @Virgo, "Is it possible that the incoming particles, in a scattering experiment, have a superposition state of up and down spin?: NOT in the case considered in scattering studies. $\endgroup$ – Arnaldo Maccarone Nov 18 '17 at 22:31
  • $\begingroup$ @ArnaldoMaccarone & Virgo: Thank you, but why is the case of a superposed spin state not considered in scattering studies? How do experimentalists guarantee that the uncertainty is classical? $\endgroup$ – MOKO Nov 19 '17 at 3:31
  • $\begingroup$ The incoming particles are prepared by the experimenter which will involves some sort of interaction with the environment and the resulting decoherence causes the particles to be in a non mixed state for the spin. $\endgroup$ – Virgo Nov 19 '17 at 3:42

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