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Spin represents polarization for other vector bosons as well. For fermions, the picture is less clear. However whether this holds for free electron is ambiguous, since for an electron, S 2 is constant, and therefore it is a matter of interpretation whether the Hamiltonian includes such a term.
Nevertheless, spin appears in the Dirac equation , and thus the relativistic Hamiltonian of the electron, treated as a Dirac field , can be interpreted as including a dependence in the spin S.
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. Composite particles also possess magnetic moments associated with their spin.
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. 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 .
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:. The long, arduous task of gaining search engine dominance in your niche just got a whole lot easier!
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