Elementary particles have intrinsic angular momentum {spin, particle}| {particle, spin} {intrinsic angular momentum}. Spin conserves energy, momentum, and angular momentum.
axis
Particles always travel at light speed along a space-time motion line. Spin axis is parallel to motion line and is either counter-clockwise or clockwise around that space-time momentum vector.
classical mechanics
In classical mechanics, spin has linear continuous projections onto other axes (and orthogonal axes have no spin components). For example, if object spins around z-axis, observers can measure spin around xz-axis and yz-axis, but spin around x-axis and y-axis (both orthogonal to z-axis) is zero.
Fundamental particles are points (or strings or loops with negligible radius), and some have no mass, so fundamental-particle intrinsic angular momentum is not due to mass rotating at a distance around an axis. Classical mechanics cannot account for elementary-particle spin.
quantum mechanics
Elementary-particle spin is quantum-mechanical and special relativistic. To reconcile quantum mechanics and special relativity, quantum-mechanical-wavefunction components are matrices, not just numbers. Matrices have transformations that are equivalent to spin angular momentum. Reconciling quantum mechanics and general relativity requires that momentum (energy) and position (time) affect each other, so matrices have complex-number elements.
In quantum mechanics, observers can measure spin around any axis. Measurement of elementary-particle spin around any axis finds that spin is an angular-momentum quantum unit, either clockwise or counterclockwise around axis. For example, measuring independent-electron intrinsic angular momentum finds spin equals (0.5 * h) / (2 * pi), where h is Planck constant, which is 1/2 angular-momentum quantum unit. (Electron spin cannot be zero, because electrons have mass.) Spin counterclockwise around motion axis adds 1/2 angular momentum unit, so spin is +1/2. Spin clockwise around motion axis subtracts 1/2 angular-momentum unit, so spin is -1/2.
Measuring independent-photon intrinsic angular momentum finds spin equals (0.5 * h) / pi, where h is Planck constant, which is 1 angular-momentum quantum unit. (Photon spin cannot be zero, because photons have energy.) Spin counterclockwise around motion axis adds 1 angular-momentum unit, so spin is +1. Spin clockwise around motion axis subtracts 1 angular-momentum unit, so spin is -1.
spin: vectors and spinors
Real-number vectors have magnitude, one direction (component), and one orientation (in that direction): (a). Rotating real-number vectors 360 degrees makes the same vector, because vector direction and orientation return to original direction and orientation. Spinning real-number vectors any number of degrees makes the same vector, because vectors have no extensions in perpendicular directions. For example, turning a straight line around its axis keeps the same shape.
Complex-number vectors have magnitude, one direction (in local two-dimensional space), and one orientation (in that direction): (a + b*i). Rotating complex-number vectors 360 degrees makes the same vector, because vector direction and orientation return to original direction and orientation. Spinning complex-number vectors any number of degrees makes the same vector, because vectors have no extensions in perpendicular directions.
Spinors have two complex-number (or quaternion) components: (a + b*i, c + d*i). Spinors have magnitude, two directions, and one orientation that depends on which component goes first. Rotating spinors 360 degrees makes original direction but opposite orientation, like rotating around a Möbius strip, because parity changes. Spinor rotation differs from vector rotation because spinor rotation has phase effects. Spinning spinors any number of degrees makes a different spinor, because spinors have extensions in perpendicular directions.
spin: rotation
Fermion odd-half-integer-spin particles have different statistics than boson integer-spin particles. For bosons, spin and rotation are independent and add. For fermions, spin and rotation are dependent and multiply.
spin: symmetries
Elementary-particle intrinsic angular momentum is about wavefunction symmetries.
Spin-0 particles are scalars (not vectors). Scalars have no direction and so have same physics under any rotation. Because intrinsic angular momentum is zero, clockwise and counterclockwise have no meaning. Spheres have all symmetries: any-degree rotational symmetry, mirror symmetry, radial symmetry, and inversion symmetry. Turning a sphere through any angle, reflecting it through any plane through any diameter, and spinning around any axis results in the same shape and behavior. Around any axis and orientation, observers see no net spin, so spin-rotation interaction is zero. See Figure 1.
Spin-1 particles are vectors, with one symmetry axis. Spin-rotation interaction is non-zero, so observers see opposite spin (anti-symmetry) after 180-degree rotation. Turning a clockwise spinning sphere upside down reverses its orientation and changes clockwise to counterclockwise. Vectors have same physics under 360-degree (and 720-degree, 1080-degree, and so on) rotation (360-degree rotational symmetry). Turning the sphere upside down again puts it back to original orientation and clockwise spin. See Figure 2.
Spin-2 particles are tensors, with two symmetry axes. Spin 2 particles have mirror symmetry. Spin 2 has 90-degree anti-symmetry. Turning the sphere to right angle interchanges axes, so one axis keeps clockwise motion and one axis changes from clockwise to counter-clockwise, reversing the orientation. Two spin-rotation interactions are non-zero but symmetric, so flipping plane over returns system to same spin-rotation interactions. Spin-2 particles have same physics under 180-degree (and 360-degree, 540-degree, 720-degree, and so on) rotation. Turning a sphere spinning clockwise around an axis and clockwise around a perpendicular axis upside down changes clockwise to counterclockwise around both axes but also reverses both axes, so the sphere returns to its original state. See Figure 3.
Spin-1/2 particles are vectors, with two axes sharing one symmetry. Because they share one symmetry, spin-1/2 particles have different spin-rotation interactions than vector bosons, which have no shared symmetry and so spin 1. Spin-rotation interaction is perpendicular at 180-degree rotation, reversed at 360-degree rotation, and opposite perpendicular at 540-degree rotation, and original at 720-degree rotation. Spin 1/2 particles have 360-degree anti-symmetry, like rotating around a Möbius strip, changing parity. Turning a sphere spinning clockwise around an axis, clockwise around a perpendicular axis, and clockwise around a second perpendicular axis completely around changes clockwise to counterclockwise around two axes and reverses both axes, but changes clockwise to counterclockwise around the third axis, which has the same orientation, so the sphere reverses orientation. Spin 1/2 has 720-degree rotational symmetry. Turning the sphere completely around again changes clockwise to counterclockwise around two axes and reverses both axes, but changes counterclockwise to clockwise around the third axis, which has the same orientation, so the sphere returns to original state. See Figure 4.
spin: speculation
Perhaps, elementary-particle intrinsic angular momentum is imaginary-number mass rotating at imaginary-number radius around particle axis, through imaginary-number angle with imaginary-number angular velocity, perhaps through imaginary-number time. Multiplying imaginary numbers results in positive real-number momentum and energy. Hyperbolas have imaginary-number radii, because they have negative curvature. Hyperbolic-curve angles are imaginary-number angles: cos(i*A) = cosh(A) and e^A = cosh(A) + sinh(A), where A is real-number angle. Higgs field has imaginary mass. Imaginary-number time rotations make special-relativity Lorentz transformations. Using imaginary-number time can establish absolute general-relativity space-time.
spin: bosons and fermions
At high concentration and/or low temperature, with Heisenberg uncertainty, for thermal-equilibrium non-interacting bosons, exchange of two particles does not change wavefunction (Bose-Einstein statistics), because particle wavefunction product is commutative (symmetric rank-two tensor): f(a) * f(b) - f(b) * f(a). Combining two spins returns the system to original orientation: f(a) * f(b) = ((-1)^(2*spin)) * (f(b) * f(a)), where spin = +1 or -1. Relativistically applying a rotation operator in imaginary time to integer spin particles results in no Pauli exclusion principle. Bosons are indistinguishable. Only system states matter. It is incorrect to talk about first one and second one, or particle 1 and particle 2. Many bosons can have same energy, momentum, and angular momentum.
At high concentration and/or low temperature, with Heisenberg uncertainty, for thermal-equilibrium non-interacting fermions, exchange of two particles changes wavefunction (Fermi-Dirac statistics), because particle wavefunction product is anti-commutative (anti-symmetric rank-two tensor): f(a) * f(b) + f(b) * f(a). Combining two spins takes the system to opposite orientation: f(a) * f(b) = ((-1)^(2*spin)) * (f(b) * f(a)), where spin = +1/2 or -1/2. Relativistically applying a rotation operator in imaginary time to half-integer spin particles results in Pauli exclusion principle. Fermions are distinguishable. Only system states matter. It is correct to talk about first one and second one, or particle 1 and particle 2. Two particles can have same energy but must have different momentum and/or angular momentum.
Note: At low concentration and/or high temperature, without Heisenberg uncertainty, thermal-equilibrium non-interacting particles have Maxwell-Boltzmann statistics. Exchange of two particles does not matter, because wavefunction has no effect. Particles can have same energy and same or different momentum and angular momentum.
spin: measurement
To measure spin, experimenters must establish a spatial axis, and then measure angular momentum around that axis. (Experimenters cannot know electron trajectories, because electrons have wavefunctions.) Around any chosen axis, instruments measure spin as exactly +1/2 unit or exactly -1/2 unit. By uncertainty principle, instruments measuring spin simultaneously around axes perpendicular to that axis get +1/2 unit or -1/2 unit with equal probability, meaning that those spin measurements have 100% uncertainty.
Instruments cannot measure spin when two electrons are interacting, because system then includes measuring apparatus. Instruments measure after particle creation or interaction. After particle creation or interaction, instruments decohere wavefunction and so destroy particle system and make particles independent.
spin: measurement angle
For electrons (spin 1/2), if measuring axis is at angle A to a clockwise spin-vector (spin -1/2), the probability that the measurement will be spin -1/2 is (cos(A/2))^2. Perhaps, because spin-vector has two axes but shares one symmetry, it is like the spin-vector projects onto an angle A/2 axis as cos(A/2), and the angle A/2 axis vector projects onto the angle A measuring axis as cos(A/2), so the net projection is (cos(A/2))^2.
If a zero-spin state emits entangled electrons in opposite directions (conserving momentum and angular momentum), and one direction is measured at angle A and the other at angle B (with angle difference C), the both-same-spin probability is (sin(C/2))^2, and the each-opposite-spin probability is (cos(C/2))^2.
For photons (spin 1), if measuring axis is at angle A to a clockwise spin-vector (spin -1), the probability that the measurement will be spin -1 is (cos(A))^2. Perhaps, because spin-vector has one axis, it is like the spin-vector projects onto an angle A axis as cos(A) twice, so the net projection is (cos(A))^2.
If a zero-spin state emits entangled photons in opposite directions (conserving momentum and angular momentum), and one direction is measured at angle A and the other at angle B (with angle difference C), the both-same-spin probability is (sin(C))^2, and the each-opposite-spin probability is (cos(C))^2.
orbitals
Orbitals with two electrons typically have one electron with positive spin and one electron with negative spin {anti-symmetric spin state}, so net spin angular momentum is zero, and ground-state orbital is symmetric. In orbitals, paired electron spins {spin pair} cancel magnetic fields.
Outside energy can add spin angular momentum. The first excited orbital state has two electrons with positive spin or two electrons with negative spin {symmetric spin state}. Net spin angular momentum is 1, and excited-state orbital is anti-symmetric.
In orbitals, two electrons have probability 0.25 to have total spin 0 and 0.75 to have total spin 1.
In different orbitals, electrons can have same lower-energy spins. Two electrons enter two different orbitals before going into same orbital, because electrostatic repulsions are greater in energy than magnetic interactions, energy differences between orbitals are small, and repulsions between electrons in different orbitals are smaller than repulsions in same orbital.
Electron has spin and can precess {spin dragging}| or move in electric fields.
Low-temperature materials can behave like ice {spin ice}|. Magnetic poles can become unaligned.
Atom electrons have coupling {spin-orbit coupling} {Russell-Sanders coupling} {jj coupling} between orbit and spin magnetic fields.
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Date Modified: 2022.0225