Spinprojektion

Spinprojektion, or spin projection, is a concept in quantum mechanics describing the value of a particle's spin angular momentum projected along a specified measurement axis. For a system with spin quantum number s, the spin-projection operator along the measurement axis has eigenvalues m hbar, where m runs from -s to +s in integer steps. In particular, for a spin-1/2 particle the possible projections are ± hbar/2, often denoted as up and down along the chosen axis.

Measurement and probabilities: The outcome is probabilistic, governed by the Born rule. If the system is in state |ψ>, the probability to obtain the eigenvalue m hbar is |<m|ψ>|^2, where |m> denotes the eigenstate of the projection operator along that axis. After measurement, the state collapses to the corresponding eigenstate.

Rotation and representation: Spin projections along different axes are related by unitary rotations. The spin state can be rotated by U(n_hat, θ) = exp(-i θ n_hat · S / hbar). In the Sz basis, a general state is α|up>_z + β|down>_z, and probabilities along z are |α|^2 and |β|^2. Higher spins have more projection outcomes m ∈ {-s, ..., s} with corresponding eigenstates.

Experimental and practical context: The concept is central to the Stern-Gerlach experiment, magnetic resonance, and quantum information, where spin projections define qubit states and readout. Projection measurements are described by projection operators P_m = |m><m|.

See also: spin, angular momentum, projection operator, Stern-Gerlach, quantum measurement.