spintoestanden

Spintoestanden, or spin states, refer to the quantum states that describe the intrinsic angular momentum of particles such as electrons, protons, and neutrons. For a spin-1/2 particle, the spin can be measured along any axis, with eigenstates of the S_z operator commonly written as |↑⟩ and |↓⟩. A general spin state is a linear combination α|↑⟩ + β|↓⟩, with complex coefficients satisfying |α|^2 + |β|^2 = 1. The Bloch sphere provides a geometric representation of such states, where the expectation value of the spin points in the direction of a Bloch vector. For higher spin S, there are 2S+1 eigenstates, labeled by m = -S, ..., S.

Measurement of spin yields discrete outcomes corresponding to projections along a chosen axis, with probabilities given by the Born rule. When considering multiple spins, the joint state lives in a tensor product space, and the combined system can exhibit entanglement if it cannot be written as a simple product of individual states. A classic example for two spin-1/2 particles is the singlet state and the triplet states, which illustrate nonclassical correlations.

Spin operators S_x, S_y, and S_z obey the angular momentum algebra [S_i, S_j] = iħ ε_ijk S_k. For spin-1/2, these operators take the form S_i = (ħ/2) σ_i, where σ_i are the Pauli matrices. Unitary evolution of spin states is described by U = exp(-i θ n·S / ħ). Magnetic fields couple to the magnetic moment proportional to S, causing phenomena such as Larmor precession.

Spintoestanden are central in many areas, including quantum information as qubits, magnetic resonance techniques, and atomic and condensed-matter physics, where coherence and relaxation define spin dynamics.