When Humpty-Dumpty had his great fall nobody could put him together again. A vastly more moderate challenge is to reunite the two partial beams of a Stern-Gerlach apparatus with such precision that the original spin state is recovered. Nevertheless, as we demonstrate, a substantial loss of spin coherence always occurs, unless the experimenter is able to control the magnetic field's inhomogeneity with an accuracy of at least one part in 105.

A unitary transformation replaces the given description of a quantum system by an equivalent one. It is observed, however, that not all members of a set of unitarily equivalent Hamilton operators are equally well suited for identifying the corresponding classical systems. A criterion is proposed for recognizing the privileged representatives of the set. A few explicit examples are reported that show the criterion at work.