a particle in which the probabilities of detecting some state are affected by the rotation of the detector. When the system is rotated through 360°, the observed output and physics are the same as initially but the amplitudes are changed for a spin- particle by a factor of −1 or a phase shift of half of 360°. When the probabilities are calculated, the −1 is squared, (−1)2 = 1, so the predicted physics is the same as in the starting position. Also, in a spin- particle there are only two spin states and the amplitudes for both change by the same −1 factor, so the interference effects are identical, unlike the case for higher spins. The complex probability amplitudes are something of a theoretical construct which cannot be directly observed. 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 from 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. In terms of more direct evidence, physical effects of the difference between the rotation of a spin- particle by 360° as compared with 720° have been experimentally observed in classic experiments in neutron interferometry. In particular, if a beam of spin-oriented spin- particles is split, and just one of the beams is rotated about the axis of its direction of motion and then recombined with the original beam, different interference effects are observed depending on the angle of rotation. In the case of rotation by 360°, cancellation effects are observed, whereas in the case of rotation by 720°, the beams are mutually reinforcing. NRQM (non-relativistic quantum mechanics) The quantum state of a spin- particle can be described by a two-component complex-valued vector called a spinor. Observable states of the particle are then found by the spin operators Sx, Sy, and Sz, and the total spin operator S. Observables When spinors are used to describe the quantum states, the three spin operators (Sx, Sy, Sz,) can be described by 2 × 2 matrices called the Pauli matrices whose eigenvalues are . For