concept

spin

Spin is an intrinsic form of angular momentum carried by elementary particles, electrons, and nuclei.
16 related articles · 5 formulas · 1 who worked here

Description

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.

History

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.

How it works

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).

Fun fact

An electron behaves as if it's spinning, but it really isn't – it's just its inherent property, like eye color.

Concept graph →

Formulas

$$ rac{m_1 oldsymbol{\chi}_1 oldsymbol{\cdot} oldsymbol{L} + m_2 oldsymbol{\chi}_2 oldsymbol{\cdot} oldsymbol{L}}{m_1 + m_2}$$
Weighted average of spin projections onto the orbital angular momentum.
What is in the formula
m_1variablemass of the first compact object — mass kilogram
m_2variablemass of the second compact object — mass kilogram
\boldsymbol{\chi}_1variabledimensionless spin vector of the first object — dimensionless spin
\boldsymbol{\chi}_2variabledimensionless spin vector of the second object — dimensionless spin
\boldsymbol{L}variableunit vector along orbital angular momentum — unit vector
\cdotoperatorcomputes the scalar projection of one vector onto another — dot product
χ_{\rmeff}=\frac{m_1χ_1\cosθ_1+m_2χ_2\cosθ_2}{m_1+m_2} · 2607.00565v1
χ_{\rmeff}\simeq\frac{m_{\rmbh}}{m_{\rmns}+m_{\rmbh}}χ_{\rmbh}\cosθ_{\rmbh} · 2607.16136v2
$$\cos i = \frac{b}{a}$$
Cosine of inclination angle equals minor axis divided by major axis.
What is in the formula
ivariableinclination angle of the orbital plane — inclination angle radian
bvariableminor axis of the projected orbit — minor axis metre
avariablemajor axis of the projected orbit — major axis metre
cosoperatorcosine function — cosine function
\cosi=\frac{b}{a} · 2607.05963
$$I = k M R^2$$
Moment of inertia for spherical objects with mass distribution.
What is in the formula
IvariableMoment of inertia — moment of inertia kilogram metre squared
kvariableDimensionless shape factor
MvariableMass of the sphere — mass kilogram
RvariableRadius of the sphere — radius metre
i\approx\frac{2}{5}(1+y)mr^2 · 2607.17609v1
$$H = -J \sum_{\langle ij \rangle} s_i s_j$$
Energy of spins interacting with nearest neighbors.
What is in the formula
HvariableHamiltonian (total energy) of the spin system — energy joule
s_ivariableSpin variable at site i (typically ±1) — spin
s_jvariableSpin variable at site j (typically ±1) — spin
JconstantStrength of the interaction between adjacent spins — exchange coupling = varies joule
∑_{⟨ij⟩}operatorSummation over all unordered pairs of adjacent lattice sites — sum over nearest neighbors
e=-j\sum_{\langleij\rangle}s_is_j · 2607.10285
$$H = -\sum_{\langle i,j \rangle} J_{ij} s_i s_j$$
Hamiltonian for Ising spin glass with random couplings.
What is in the formula
HvariableHamiltonian of the Ising spin glass — hamiltonian joule
J_{ij}variableRandom exchange coupling between spins i and j — exchange coupling joule
s_ivariableIsing spin variable at site i — spin
s_jvariableIsing spin variable at site j — spin
\sum_{\langle i,j \roperatorSummation over all nearest-neighbor pairs on the lattice — summation

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