electronspin

Electron spin is an intrinsic form of angular momentum carried by electrons, a fundamental quantum property with no classical counterpart. It has two possible projections along any measurement axis, commonly described as up and down.

Mathematically, spin is described by a two-component Pauli spinor and a set of spin operators S = (ħ/2) σ, where σ are the Pauli matrices. The state can be written as |ψ> = α|↑> + β|↓>, with complex coefficients α and β and |α|^2 + |β|^2 = 1. In a magnetic field, spin interacts with the field through a magnetic moment μ = -g_s μ_B S/ħ, yielding an energy E = -μ·B. For an eigenstate along z, this gives E = ±(g_s μ_B/2) B. The electron g-factor is approximately g_s ≈ 2, with small corrections.

Spin is not due to literal spinning in space; it is a quantum degree of freedom that produces a magnetic moment and quantized measurement outcomes. The quantization was famously demonstrated in Stern-Gerlach experiments. Spin states can be superposed and entangled, and the total wavefunction of multi-electron systems must be antisymmetric under particle exchange, reflecting the fermionic nature of electrons.

In research and technology, electron spin underpins ESR/EPR spectroscopy, spintronics, and quantum information processing. Spin qubits are realized in systems such as quantum dots, donor spins in silicon, and defects in solids, where coherent control and readout of spin states enable quantum operations.