Spin-1/2
Unlike in more complicated quantum mechanical systems, the spin of a spin-1/2 particle can be expressed as a linear combination of just two eigenstates, or eigenspinors.
[4] Mathematically, quantum mechanical spin is not described by a vector as in classical angular momentum.
In mathematical terms, the quantum Hilbert space carries a projective representation of the rotation group SO(3).
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-1/2 particle by a factor of −1 or a phase shift of half of 360°.
In terms of more direct evidence, physical effects of the difference between the rotation of a spin-1/2 particle by 360° as compared with 720° have been experimentally observed in classic experiments[5] in neutron interferometry.
[5] The quantum state of a spin-1/2 particle can be described by a two-component complex-valued vector called a spinor.
The two eigenvalues of Sz, ±ħ/2, then correspond to the following eigenspinors: These vectors form a complete basis for the Hilbert space describing the spin-1/2 particle.
[citation needed] As a consequence of the four-dimensional nature of space-time in relativity, relativistic quantum mechanics uses 4×4 matrices to describe spin operators and observables.
[citation needed] When physicist Paul Dirac tried to modify the Schrödinger equation so that it was consistent with Einstein's theory of relativity, he found it was only possible by including matrices in the resulting Dirac equation, implying the wave must have multiple components leading to spin.
[7] The 4π spinor rotation was experimentally verified using neutron interferometry in 1974, by Helmut Rauch and collaborators,[8] after being suggested by Yakir Aharonov and Leonard Susskind in 1967.
