If the probability amplitudes rotated by the same amount as the detector, then they would have changed by a factor of −1 when the equipment was rotated by 180° which when squared would predict the same output as at the start, but experiments show this to be wrong. If the detector is rotated by 180°, the result with spin- particles can be different to what it would be if not rotated, hence the factor of a half is necessary to make the predictions of the theory match the experiments.