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Spin - Spin (1976)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: [14] [15] [16].

New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments. Experimental results have put the neutrino magnetic moment at less than 1.

On the other hand elementary particles with spin but without electric charge, such as a photon or a Z boson, do not have a magnetic moment.

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 for a spin-s particle measured along any direction can only take on the values [18].

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.

The Plate trick and Möbius strip give non-quantum analogies. 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.

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 is 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. A Journal On Joy. Discovering your own strength and pushing yourself to the limit.

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