The Spin

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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.

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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.

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The minimally extended Standard Model that takes into account non-zero neutrino masses predicts neutrino magnetic moments of: [13] [14] [15]. 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.


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Experimental results have put the neutrino magnetic moment at less than 1. 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.

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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 [17].

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: s x , s y and s z cannot possess simultaneous definite values, because of a quantum uncertainty relation between them.

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. For example, one can exert a kind of " torque " on an electron by putting it in a magnetic field the field acts upon the electron's intrinsic magnetic dipole moment —see the following section. 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.

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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.

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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. The most convenient quantum mechanical description of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis.

Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It's clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal up to phase to the matrix representing rotation AB.

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. Soon the terminology 'spin' was used to describe this apparent rotation of subatomic particles. It is analogous to the spin of a planet in that it gives a particle angular momentum and a tiny magnetic field called a magnetic moment. Based on the known sizes of subatomic particles, however, the surfaces of charged particles would have to be moving faster than the speed of light in order to produce the measured magnetic moments. Furthermore, spin is quantized, meaning that only certain discrete spins are allowed. This situation creates all sorts of complications that make spin one of the more challenging aspects of quantum mechanics.

Spin is likewise an essential consideration in all interactions among subatomic particles, whether in high-energy particle beams, low-temperature fluids or the tenuous flow of particles from the sun known as the solar wind. Indeed, many if not most physical processes, ranging from the smallest nuclear scales to the largest astrophysical distances, depend greatly on interactions of subatomic particles and the spins of those particles. Stenger, professor of physics at the University of Hawaii at Manoa, offers another, more technical perspective: "Spin is the total angular momentum, or intrinsic angular momentum, of a body.

The spins of elementary particles are analogous to the spins of macroscopic bodies. In fact, the spin of a planet is the sum of the spins and the orbital angular momenta of all its elementary particles. So are the spins of other composite objects such as atoms, atomic nuclei and protons which are made of quarks. In quantum mechanics, angular momenta are discrete, quantized in units of Planck's constant divided by 4 pi.

Niels Bohr proposed that angular momentum is quantized in and used this to explain the line spectrum of hydrogen. These particles are all imagined as pointlike, so you might wonder how they can have spins. This is confirmed by the blue and red dashed lines that display the Bethe-Ansatz solutions for temperatures 0.

The importance of corrections to scaling increases from A to D , spoiling a complete scaling collapse. The dashed lines show the universal divergences of Eq. The critical is plotted as solid lines in Fig.


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