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This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three x -, y -, z -axis directions.

In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity.

Since the Pauli matrices do not commute , measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the x -axis, and we then measure the spin along the y -axis, we have invalidated our previous knowledge of the x -axis spin.

This can be seen from the property of the eigenvectors i. This implies that the original measurement of the spin along the x-axis is no longer valid, since the spin along the x -axis will now be measured to have either eigenvalue with equal probability.

By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large s , can be calculated using this spin operator and ladder operators.

The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis. Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G n is defined to consist of all n -fold tensor products of Pauli matrices.

The analog formula of Euler's formula in terms of the Pauli matrices :. For example, see the isotopes of bismuth in which the List of isotopes includes the column Nuclear spin and parity.

Spin has important theoretical implications and practical applications. Well-established direct applications of spin include:. Electron spin plays an important role in magnetism , with applications for instance in computer memories.

The manipulation of nuclear spin by radiofrequency waves nuclear magnetic resonance is important in chemical spectroscopy and medical imaging.

Spin-orbit coupling leads to the fine structure of atomic spectra, which is used in atomic clocks and in the modern definition of the second.

Precise measurements of the g -factor of the electron have played an important role in the development and verification of quantum electrodynamics.

Photon spin is associated with the polarization of light photon polarization. An emerging application of spin is as a binary information carrier in spin transistors.

The original concept, proposed in , is known as Datta-Das spin transistor. The manipulation of spin in dilute magnetic semiconductor materials , such as metal-doped ZnO or TiO 2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.

There are many indirect applications and manifestations of spin and the associated Pauli exclusion principle , starting with the periodic table of chemistry.

Spin was first discovered in the context of the emission spectrum of alkali metals. In , Wolfgang Pauli introduced what he called a "two-valuedness not describable classically" [24] associated with the electron in the outermost shell.

This allowed him to formulate the Pauli exclusion principle , stating that no two electrons can have the same quantum state in the same quantum system.

The physical interpretation of Pauli's "degree of freedom" was initially unknown. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum.

This would violate the theory of relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea.

Under the advice of Paul Ehrenfest , they published their results. This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.

Mathematically speaking, a fiber bundle description is needed. The tangent bundle effect is additive and relativistic; that is, it vanishes if c goes to infinity.

It is one half of the value obtained without regard for the tangent space orientation, but with opposite sign. Thus the combined effect differs from the latter by a factor two Thomas precession , known to Ludwik Silberstein in Despite his initial objections, Pauli formalized the theory of spin in , using the modern theory of quantum mechanics invented by Schrödinger and Heisenberg.

He pioneered the use of Pauli matrices as a representation of the spin operators, and introduced a two-component spinor wave-function.

Pauli's theory of spin was non-relativistic. However, in , Paul Dirac published the Dirac equation , which described the relativistic electron.

In the Dirac equation, a four-component spinor known as a " Dirac spinor " was used for the electron wave-function. Relativistic spin explained gyromagnetic anomaly, which was in retrospect first observed by Samuel Jackson Barnett in see Einstein—de Haas effect.

In , Pauli proved the spin-statistics theorem , which states that fermions have half-integer spin and bosons have integer spin.

In retrospect, the first direct experimental evidence of the electron spin was the Stern—Gerlach experiment of However, the correct explanation of this experiment was only given in From Wikipedia, the free encyclopedia.

Intrinsic form of angular momentum as a property of quantum particles. This article is about spin in quantum mechanics. For rotation in classical mechanics, see Angular momentum.

Elementary particles of the Standard Model. Main article: Spin quantum number. Main article: spin—statistics theorem.

Main article: Spin magnetic moment. Further information: Angular momentum operator. Main article: Pauli matrices.

See also: Symmetry in quantum mechanics. Click "show" at right to see a proof or "hide" to hide it. This section needs additional citations for verification.

Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.

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