Electron Spin
Electron spin is an intrinsic quantum mechanical property of electrons (and other fermions) that represents a form of angular momentum with no classical analogue — the electron does not physically rotate, but behaves as if it does, with a fixed magnitude of angular momentum. Electrons have a spin quantum number s = 1/2, giving two possible spin states: spin-up (mₛ = +1/2) and spin-down (mₛ = −1/2). Electron spin is responsible for the Pauli Exclusion Principle, magnetic properties of atoms (paramagnetism and diamagnetism), and is the basis for technologies such as MRI and spintronics.
Key Formula
|S| = ℏ × √(s(s+1)) = (√3/2) × ℏ
LaTeX: |S| = \hbar\sqrt{s(s+1)} = \frac{\sqrt{3}}{2}\hbar
| Symbol | Meaning | Unit |
|---|---|---|
| |S| | Magnitude of spin angular momentum | J·s |
| ℏ | Reduced Planck's constant (1.055 × 10⁻³⁴) | J·s |
| s | Spin quantum number (= 1/2 for electrons) | Dimensionless |
Worked Example
Problem
Calculate the magnitude of the spin angular momentum of an electron.
Solution
Step 1: Use the spin angular momentum formula. |S| = ℏ√(s(s+1)) Step 2: Substitute s = 1/2 for an electron. |S| = ℏ√((1/2)(1/2 + 1)) |S| = ℏ√((1/2)(3/2)) |S| = ℏ√(3/4) |S| = ℏ × (√3)/2 Step 3: Substitute ℏ = 1.055 × 10⁻³⁴ J·s. |S| = 1.055 × 10⁻³⁴ × 0.8660 |S| = 9.133 × 10⁻³⁵ J·s
Answer
|S| ≈ 9.13 × 10⁻³⁵ J·s (fixed magnitude of electron spin angular momentum)
Comparison of Spin Properties for Common Particles
| Particle | Spin (s) | mₛ values | Classification | Example Property |
|---|---|---|---|---|
| Electron | 1/2 | +1/2, −1/2 | Fermion | Pauli exclusion applies |
| Proton | 1/2 | +1/2, −1/2 | Fermion | Nuclear magnetic resonance |
| Photon | 1 | +1, 0, −1 | Boson | Circular polarisation |
| Higgs boson | 0 | 0 | Boson | Scalar field |
| Helium-4 nucleus | 0 | 0 | Boson | Bose-Einstein condensate |
Interactive Tools
PhET Stern-Gerlach Experiment
Simulate the Stern-Gerlach experiment demonstrating electron spin quantisation
Open ToolKhan Academy – Electron Spin
Explanation of the spin quantum number and its role in electron configurations
Open ToolBrilliant – Spin Angular Momentum
Mathematical and physical treatment of quantum spin with practice problems
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Quantum Number
Quantum numbers are discrete integer or half-integer values that characterise the quantum state of an electron (or other quantum particle) in an atom, arising naturally as solutions to the Schrödinger equation. The four quantum numbers for electrons — principal (n), azimuthal (l), magnetic (mₗ), and spin (mₛ) — together uniquely specify the quantum state of each electron in accordance with the Pauli Exclusion Principle. They determine the energy, shape, spatial orientation, and spin of atomic orbitals, providing the foundation for the periodic table and chemical bonding.
Atomic Orbital
An atomic orbital is a mathematical function describing the wave-like behaviour and probable location of an electron in an atom, representing a region of space where there is a high probability (typically 90–95%) of finding the electron. Orbitals are characterised by three quantum numbers (n, l, mₗ) and have distinct shapes: s-orbitals are spherical, p-orbitals are dumbbell-shaped, and d- and f-orbitals have more complex geometries. Atomic orbitals form the basis for understanding electron configurations, chemical bonding, and molecular orbital theory.
Quantum Mechanics
Quantum mechanics is the fundamental theory of physics that describes the behaviour of matter and energy at the scale of atoms and subatomic particles, where classical Newtonian mechanics breaks down. It introduces concepts such as quantisation of energy, wave-particle duality, and the probabilistic nature of physical observables. Quantum mechanics underpins modern technologies including semiconductors, lasers, MRI machines, and quantum computing.
The term "spin" was introduced by George Uhlenbeck and Samuel Goudsmit in 1925 to describe the intrinsic angular momentum of the electron, by analogy with a spinning top. It derives from Old English "spinnan" (to spin). The theoretical foundation was provided by Paul Dirac's relativistic quantum equation in 1928.