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Inquantum mechanics,spin is an intrinsic property of allelementary particles. All knownfermions, the particles that constitute ordinary matter, have a spin of⁠1/2⁠.^[1][2][3] The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of⁠1/2⁠ means that the particle must be rotated by two fullturns (through 720°) before it has the same configuration as when it started.

Particles with netspin⁠1/2⁠ include theproton,neutron,electron,neutrino, andquarks. The dynamics of spin-⁠1/2⁠ objects cannot be accurately described usingclassical physics; they are among the simplest systems whose description requiresquantum mechanics. As such, the study of the behavior of spin-⁠1/2⁠ systems forms a central part ofquantum mechanics.

The necessity of introducing half-integerspin goes back experimentally to the results of theStern–Gerlach experiment. A beam of atoms is run through a strong heterogeneous magnetic field, causing it to split intoN parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two—theground state therefore could not be an integer, because even if the intrinsic angular momentum of the atoms were the smallest (non-zero) integer possible, 1, the beam would be split into 3 parts, corresponding to atoms withL_z = −1, +1, and 0, with 0 simply being the value known to come between −1 and +1 while also being a whole-integer itself, and thus a valid quantized spin number in this case. The existence of this hypothetical "extra step" between the two polarized quantum states would necessitate a third quantum state, a third beam, which is not observed in the experiment. The conclusion was that silver atoms had net intrinsic angular momentum of⁠1/2⁠.^[1]

Spin-⁠1/2⁠ objects are allfermions (a fact explained by thespin–statistics theorem) and satisfy thePauli exclusion principle. Spin-⁠1/2⁠ particles can have a permanentmagnetic moment along the direction of their spin, and this magnetic moment gives rise toelectromagnetic interactions that depend on the spin. One such effect that was important in the discovery of spin is theZeeman effect—the splitting of a spectral line into several components in the presence of a static magnetic field.

Unlike in more complicated quantum mechanical systems, the spin of a spin-⁠1/2⁠ particle can be expressed as alinear combination of just twoeigenstates, oreigenspinors. These are traditionally labeled spin up and spin down. Because of this, the quantum-mechanical spinoperators can be represented as simple 2 × 2matrices. These matrices are called thePauli matrices.

Creation and annihilation operators can be constructed for spin-⁠1/2⁠ objects; these obey the samecommutation relations as otherangular momentum operators.

One consequence of thegeneralized uncertainty principle is that the spin projection operators (which measure the spin along a given direction likex,y, orz) cannot be measured simultaneously. Physically, this means that the axis about which a particle is spinning is ill-defined. A measurement of thez-component of spin destroys any information about thex- andy-components that might previously have been obtained.

A spin-⁠1/2⁠ particle is characterized by anangular momentum quantum number for spins of⁠1/2⁠. In solutions of theSchrödinger equation, angular momentum is quantized according to this number, so that the total spin angular momentum is

${\displaystyle S=\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)}\hslash =\frac{\sqrt{3}}{2}\hslash .}$

However, the observedfine structure when the electron is observed along one axis, such as thez-axis, is quantized in terms of amagnetic quantum number, which can be viewed as a quantization of avector component of this total angular momentum, which can have only the values of$\pm \frac{1}{2}\hslash$.

Note that these values for angular momentum are functions only of thereduced Planck constant (the angular momentum of anyphoton), with no dependence on mass or charge.^[4]

Mathematically, quantum mechanical spin is not described by avector as in classical angular momentum. It is described by a complex-valued vector with two components called aspinor. There are subtle differences between the behavior of spinors and vectors undercoordinate rotations, stemming from the behavior of a vector space over a complex field.

When a spinor is rotated by 360° (one full turn), it transforms to its negative, and then after a further rotation of 360°, it transforms back to its initial value again. This is because in quantum theory the state of a particle or system is represented by a complexprobability amplitude (wavefunction)${\displaystyle \psi }$, and when the system is measured, the probability of finding the system in the state${\displaystyle \psi }$ equals$\left|{\psi }^2\right|=\psi \ast \psi$, theabsolute square (square of theabsolute value) of the amplitude. In mathematical terms, the quantum Hilbert space carries aprojective representation of the rotation group SO(3).

Suppose a detector that can be rotated measures 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-⁠1/2⁠ 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-⁠1/2⁠ 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 that 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-⁠1/2⁠ 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-⁠1/2⁠ particle by 360° as compared with 720° have been experimentally observed in classic experiments^[5] in neutron interferometry. In particular, if a beam of spin-oriented spin-⁠1/2⁠ 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.^[5]

Thequantum state of a spin-⁠1/2⁠ particle can be described by a two-component complex-valued vector called aspinor. Observable states of the particle are then found by the spin operatorsS_x,S_y, andS_z, and the total spin operator S.

When spinors are used to describe the quantum states, the three spin operators (S_x,S_y,S_z) can be described by 2 × 2 matrices called the Pauli matrices whoseeigenvalues are$\pm \frac{1}{2}\hslash$.

For example, the spin projection operatorS_z affects a measurement of the spin in thez direction:

The two eigenvalues ofS_z,$\pm \frac{1}{2}\hslash$, then correspond to the following eigenspinors:

${\displaystyle {\chi }_{+}=\left[\begin{array}{c}1\\ 0\end{array}\right]=|s_z=+\frac{1}{2}⟩=|\uparrow ⟩=|0⟩}$

${\displaystyle {\chi }_{-}=\left[\begin{array}{c}0\\ 1\end{array}\right]=|s_z=-\frac{1}{2}⟩=|\downarrow ⟩=|1⟩.}$

These vectors form a complete basis for theHilbert space describing the spin-⁠1/2⁠ particle. Thus, linear combinations of these two states can represent all possible states of the spin, including in thex- andy-directions.

Theladder operators are:

Since${\displaystyle S_{\pm }=S_x\pm i{S}_y}$,^[6] it follows that$S_x=\frac{1}{2}(S_{+}+S_{-})$ and$S_y=\frac{1}{2i}(S_{+}-S_{-})$. Thus:

Their normalized eigenspinors can be found in the usual way. For$S_x$, they are:

${\displaystyle {\chi }_{+}^{(x)}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 1\end{array}\right]=|s_x=+\frac{1}{2}⟩}$

${\displaystyle {\chi }_{-}^{(x)}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -1\end{array}\right]=|s_x=-\frac{1}{2}⟩}$

For$S_y$, they are:

${\displaystyle {\chi }_{+}^{(y)}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ i\end{array}\right]=|s_y=+\frac{1}{2}⟩}$

${\displaystyle {\chi }_{-}^{(y)}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -i\end{array}\right]=|s_y=-\frac{1}{2}⟩}$

While non-relativistic quantum mechanics defines spin⁠1/2⁠ with 2 dimensions in Hilbert space with dynamics that are described in 3-dimensional space and time,relativistic quantum mechanics defines the spin with 4 dimensions in Hilbert space and dynamics described by 4-dimensional space-time.^{[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 physicistPaul Dirac tried to modify theSchrödinger equation so that it was consistent with Einstein'stheory of relativity, he found it was only possible by including matrices in the resultingDirac equation, implying the wave must have multiple components leading to spin.^[7]

The 4π spinor rotation was experimentally verified using neutron interferometry in 1974 byHelmut Rauch and collaborators,^[8] after being suggested byYakir Aharonov andLeonard Susskind in 1967.^[9]

• Projective representation

• Feynman, Richard (1963)."Volume III, Chapter 6. Spin One-Half".The Feynman Lectures on Physics.Caltech.
• Penrose, Roger (2007).The Road to Reality. Vintage Books.ISBN 978-0-679-77631-4.

• Media related toSpin-½ at Wikimedia Commons

• Rotation in three dimensions
• Quantum models

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