State reduction in quantum mechanics: A calculational example

The question is posed as to the role of the state reduction hypothesis in the passage of a molecular beam through a Stern-Gerlach apparatus of the ordinary sort, and of the sort proposed by Wigner, in which the beam is rejoined at the end of the apparatus. The effect of the magnetic field is calculated, as is the effect of a “detector” atom placed asymmetrically in the apparatus, and sensitive to the passage of the molecular beam nearby. It is shown that in the absence of the detector atom the rejoined beam is represented by a pure case density matrix. However, when the detector is present in one arm of our Stern-Gerlach apparatus, the beam is found to be in a statistical mixture approximating “reduction”, even if we never “look” at the detector atom. It is also seen that the beam density matrix is reduced to diagonal form even though it is not physically affected by the detector.