
Spin is an intrinsic form of angular momentum carried by elementary particles, electrons, and nuclei.Spin is like a tiny top that every particle has, but it doesn't actually spin. It's just an intrinsic property, similar to rotation, but without motion.
The experimental discovery of spin is linked to the Stern–Gerlach experiment (1922), where a beam of silver atoms split into two components in an inhomogeneous magnetic field, contradicting classical physics. The theoretical explanation was given by George Uhlenbeck and Samuel Goudsmit in 1925, proposing an electron’s own momentum. Later, Pauli introduced spin matrices, and Dirac in 1928 naturally incorporated spin into the relativistic equation for the electron. Subsequently, the concept of spin became key for all quantum field theory and particle physics.
Spin is described by operators obeying the commutation relations [Ŝ_i, Ŝ_j] = iħ ε_{ijk} Ŝ_k. For spin-1/2 particles, the Pauli representation with matrices σ_x, σ_y, σ_z is often used. The Hamiltonian for interaction with a magnetic field: H = –μ·B, where the magnetic moment μ = g (e/2m) S. Quantum spin correlations (entanglement) are tested in Bell's inequalities, demonstrating the nonlocality of quantum mechanics. Spin is also responsible for the fine structure of spectra and plays a central role in spin electronics (spintronics).
An electron behaves as if it's spinning, but it really isn't – it's just its inherent property, like eye color.
| m_1 | variable | mass of the first compact object — mass kilogram |
| m_2 | variable | mass of the second compact object — mass kilogram |
| \boldsymbol{\chi}_1 | variable | dimensionless spin vector of the first object — dimensionless spin |
| \boldsymbol{\chi}_2 | variable | dimensionless spin vector of the second object — dimensionless spin |
| \boldsymbol{L} | variable | unit vector along orbital angular momentum — unit vector |
| \cdot | operator | computes the scalar projection of one vector onto another — dot product |
| i | variable | inclination angle of the orbital plane — inclination angle radian |
| b | variable | minor axis of the projected orbit — minor axis metre |
| a | variable | major axis of the projected orbit — major axis metre |
| cos | operator | cosine function — cosine function |
| I | variable | Moment of inertia — moment of inertia kilogram metre squared |
| k | variable | Dimensionless shape factor |
| M | variable | Mass of the sphere — mass kilogram |
| R | variable | Radius of the sphere — radius metre |
| H | variable | Hamiltonian (total energy) of the spin system — energy joule |
| s_i | variable | Spin variable at site i (typically ±1) — spin |
| s_j | variable | Spin variable at site j (typically ±1) — spin |
| J | constant | Strength of the interaction between adjacent spins — exchange coupling = varies joule |
| ∑_{⟨ij⟩} | operator | Summation over all unordered pairs of adjacent lattice sites — sum over nearest neighbors |
| H | variable | Hamiltonian of the Ising spin glass — hamiltonian joule |
| J_{ij} | variable | Random exchange coupling between spins i and j — exchange coupling joule |
| s_i | variable | Ising spin variable at site i — spin |
| s_j | variable | Ising spin variable at site j — spin |
| \sum_{\langle i,j \r | operator | Summation over all nearest-neighbor pairs on the lattice — summation |



















