Quote:Wolfgang Pauli was the first to propose the concept of spin, but he did not name it. In 1925, Ralph Kronig, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested a physical interpretation of particles spinning around their own axis.

Quote:As the name suggests, spin was originally conceived as the rotation of a particle around some axis.

Quote:Those particles with half-integer spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer spins, such as 0, 1, 2, are known as bosons. The two families of particles obey different rules and broadly have different roles in the world around us. A key distinction between the two families is that fermions obey the Pauli exclusion principle; that is, there cannot be two identical fermions simultaneously having the same quantum numbers (meaning, roughly, having the same position, velocity and spin direction). In contrast, bosons obey the rules of Bose–Einstein statistics and have no such restriction, so they may "bunch together" even if in identical states. Also, composite particles can have spins different from the particles which comprise them. For example, a helium atom can have spin 0 and therefore can behave like a boson even though the quarks and electrons which make it up are all fermions.

This has profound practical applications:

Quote:In quantum mechanics, spin is an intrinsic property of all elementary particles. Fermions, the particles that constitute ordinary matter, have half-integer spin. Spin-½ particles constitute an important subset of such fermions. All known elementary fermions have a spin of ½

Quote:Particles having net spin ½ include the proton, neutron, electron, neutrino, and quarks. The dynamics of spin-½ objects cannot be accurately described using classical physics; they are among the simplest systems which require quantum mechanics to describe them. As such, the study of the behavior of spin-½ systems forms a central part of quantum mechanics.

Quote:Unlike in more complicated quantum mechanical systems, the spin of a spin-½ particle can be expressed as a linear combination of just two eigenstates, or eigenspinors. These are traditionally labeled spin up and spin down. Because of this the quantum mechanical spin operators can be represented as simple 2 × 2 matrices. These matrices are called the Pauli matrices.

Quote:Complex phase
A single point in space can spin continuously without becoming tangled. Notice that after a 360 degree rotation, the spiral flips between clockwise and counterclockwise orientations. It returns to its original configuration after spinning a full 720 degrees.

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

When a spinor is rotated by 360 degrees (one full turn), it becomes negative, then after a further rotation 360 degrees becomes positive again. This comes about because in quantum theory the state of a particle or system is represented by a complex probability amplitude (wavefunction) Ψ, and when the system is measured; the probability of finding the system in the state Ψ equals |Ψ|2 = Ψ*Ψ, the square of absolute value of the amplitude.

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 degrees the observed output and physics are the same as initially but the amplitudes are changed for a spin-½ particle by a factor of −1 or a phase shift of half of 360 degrees. When the probabilities are calculated the −1 is squared; (−1)2 = 1, so the predicted physics is same as in the starting position. Also in a spin-½ 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 and 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 degrees, which when squared would predict the same output as at the start but this is wrong experimentally. If the detector is rotated by 180 degrees, the result with spin-½ particles can be different to what it would be if not rotated, hence the factor of a half is necessary to make the predictions of the theory match experiment.

Quote:180 degrees of rotation isn't interesting. It reverses the direction of spin, as you would expect. And 720 degrees always reproduces the initial configuration.

Quote:Chirality /kaɪˈrælɪtiː/ is a property of asymmetry important in several branches of science. The word chirality is derived from the Greek, χειρ (kheir), "hand", a familiar chiral object.

An object or a system is chiral if it is distinguishable from its mirror image; that is, it cannot be superposed onto it. Conversely, a mirror image of an achiral object, such as a sphere, cannot be distinguished from the object. A chiral object and its mirror image are called enantiomorphs (Greek opposite forms) or, when referring to molecules, enantiomers. A non-chiral object is called achiral (sometimes also amphichiral) and can be superposed on its mirror image.

Quote:A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity. Invariance under parity by a Dirac fermion is called chiral symmetry.

An experiment on the weak decay of cobalt-60 nuclei carried out by Chien-Shiung Wu and collaborators in 1957 demonstrated that parity is not a symmetry of the universe.