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Spin is an intrinsic angular momentum associated with quantum mechanical particles. Unlike classical "spinning" objects, which derive their angular momentum from the rotation of their constituent parts, spin angular momentum is not associated with any rotating internal masses. For example, elementary particles, such as the electron, possess spin angular momentum, even though they are point particles. Also, unlike classical mechanical spinning, the spin is not described by a vector, but by a two-component object (for spin-1/2 particles): there is an observable difference in how it transforms under coordinate rotations.

Other subatomic particles, such as neutrons, which have zero electrical charge, also possess spin.

When applied to spatial rotations, the principles of quantum mechanics state that the observed values of angular momentum (which are eigenvalues of the angular momentum operator) are restricted to integer or half-integer multiples of h/2&pi. This applies to spin angular momentum as well. Furthermore, the spin-statistics theorem states that particles with integer spin correspond to bosons, and particles with half-integer spin correspond to fermions.

A rotating charged body in an inhomogenous magnetic field will experience a force. Electrons in an inhomogenous magnetic field also experience a force, and this is why people have imagined the electron as "spinning around". The observed forces vary for different electrons, and these differences are attributed to differences in spin. The spin of electrons is therefore typically measured by observing their deflection in an inhomogenous magnetic field. In accordance with the predictions of theory, only half-integer multiples of h/2π are ever observed for electrons.

## History

Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924, Wolfgang Pauli (who was possibly the most influential physicist in the theory of spin) introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can share the same quantum numbers.

The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. 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.

In the fall of that year, the same thought came to two young Dutch physicists, George Uhlenbeck and Samuel Goudsmit. Under the advice of Paul Ehrenfest, they published their results in a small paper. It met a favorable response, especially after L.H. Thomas managed to resolve a factor of two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished ones.) This discrepancy was due to the necessity to take into account 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 (i.e. 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).

Despite his initial objections to the idea, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics discovered 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 1928, 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.

In 1940, Pauli proved the spin-statistics theorem, which states that fermions have half-integer spin and bosons integer spin.

## Application

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