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On the other hand, spin has some peculiar properties that distinguish it from orbital angular momenta:. The value of s for an elementary particle depends only on the type of particle, and cannot be altered in any known way in contrast to the spin direction described below.
The spin angular momentum, S , of any physical system is quantized. The allowed values of S are. In contrast, orbital angular momentum can only take on integer values of s ; i.
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 their component particles. For example, a helium atom in the ground state has spin 0 and behaves like a boson, even though the quarks and electrons which make it up are all fermions.
The spin-statistics theorem states 1 that particles with half-integer spin fermions obey Fermi—Dirac statistics and the Pauli Exclusion Principle, and 2 that particles with integer spin bosons obey Bose—Einstein statistics, occupy "symmetric states", and thus can share quantum states.
The theorem relies on both quantum mechanics and the theory of special relativity , and this connection between spin and statistics has been called "one of the most important applications of the special relativity theory".
Particles with spin can possess a magnetic dipole moment , just like a rotating electrically charged body in classical electrodynamics.
These magnetic moments can be experimentally observed in several ways, e. For exclusively orbital rotations it would be 1 assuming that the mass and the charge occupy spheres of equal radius.
The electron, being a charged elementary particle, possesses a nonzero magnetic moment. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral.
This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks , which are electrically charged particles.
The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions. Neutrinos are both elementary and electrically neutral.
The minimally extended Standard Model that takes into account non-zero neutrino masses predicts neutrino magnetic moments of: New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments.
The measurement of neutrino magnetic moments is an active area of research. In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction, with the overall average being very near zero.
Ferromagnetic materials below their Curie temperature , however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, producing a macroscopic, non-zero magnetic field from the domain.
These are the ordinary "magnets" with which we are all familiar. In paramagnetic materials, the magnetic dipole moments of individual atoms spontaneously align with an externally applied magnetic field.
In diamagnetic materials, on the other hand, the magnetic dipole moments of individual atoms spontaneously align oppositely to any externally applied magnetic field, even if it requires energy to do so.
The study of the behavior of such " spin models " is a thriving area of research in condensed matter physics. For instance, the Ising model describes spins dipoles that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction.
These models have many interesting properties, which have led to interesting results in the theory of phase transitions. In classical mechanics, the angular momentum of a particle possesses not only a magnitude how fast the body is rotating , but also a direction either up or down on the axis of rotation of the particle.
Quantum mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum measured along any direction can only take on the values .
Conventionally the direction chosen is the z -axis:. This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation.
It turns out that the spin vector is not very useful in actual quantum mechanical calculations, because it cannot be measured directly: However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a Stern—Gerlach apparatus , the spin vector does have a well-defined experimental meaning: As a qualitative concept, the spin vector is often handy because it is easy to picture classically.
For instance, quantum mechanical spin can exhibit phenomena analogous to classical gyroscopic effects. The result is that the spin vector undergoes precession , just like a classical gyroscope.
This phenomenon is known as electron spin resonance ESR. The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance NMR spectroscopy and imaging.
Mathematically, quantum-mechanical spin states are described by vector-like objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations.
To return the particle to its exact original state, one needs a degree rotation. A spin-zero particle can only have a single quantum state, even after torque is applied.
Rotating a spin-2 particle degrees can bring it back to the same quantum state and a spin-4 particle should be rotated 90 degrees to bring it back to the same quantum state.
The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated degrees and a spin 0 particle can be imagined as sphere, which looks the same after whatever angle it is turned through.
Spin obeys commutation relations analogous to those of the orbital angular momentum:. It follows as with angular momentum that the eigenvectors of S 2 and S z expressed as kets in the total S basis are:.
The spin raising and lowering operators acting on these eigenvectors give:. But unlike orbital angular momentum the eigenvectors are not spherical harmonics.
There is also no reason to exclude half-integer values of s and m s. In addition to their other properties, all quantum mechanical particles possess an intrinsic spin though this value may be equal to zero.
One distinguishes bosons integer spin and fermions half-integer spin. The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.
For systems of N identical particles this is related to the Pauli exclusion principle , which states that by interchanges of any two of the N particles one must have.
In quantum mechanics all particles are either bosons or fermions. In some speculative relativistic quantum field theories " supersymmetric " particles also exist, where linear combinations of bosonic and fermionic components appear.
The above permutation postulate for N -particle state functions has most-important consequences in daily life, e.
As described above, quantum mechanics states that components of angular momentum measured along any direction can only take a number of discrete values.
Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. Further, rotations preserve the quantum mechanical inner product, and so should our transformation matrices:.
Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO 3. Each such representation corresponds to a representation of the covering group of SO 3 , which is SU 2.
Starting with S x. Using the spin operator commutation relations , we see that the commutators evaluate to i S y for the odd terms in the series, and to S x for all of the even terms.
Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension i. A generic rotation in 3-dimensional space can be built by compounding operators of this type using Euler angles:.
An irreducible representation of this group of operators is furnished by the Wigner D-matrix:. Recalling that a generic spin state can be written as a superposition of states with definite m , we see that if s is an integer, the values of m are all integers, and this matrix corresponds to the identity operator.
This fact is a crucial element of the proof of the spin-statistics theorem. We could try the same approach to determine the behavior of spin under general Lorentz transformations , but we would immediately discover a major obstacle.
Unlike SO 3 , the group of Lorentz transformations SO 3,1 is non-compact and therefore does not have any faithful, unitary, finite-dimensional representations.
These spinors transform under Lorentz transformations according to the law. It can be shown that the scalar product.
The corresponding normalized eigenvectors are:. Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign.
In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as sympy; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.
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