spinprojection

Spin projection, also called the projection of spin along a chosen axis, is the quantum observable given by the operator S · n, where S is the total spin angular momentum operator and n is a unit vector indicating the measurement axis. The eigenvalues of S · n are the possible spin projections along n, equal to m_s ℏ, where m_s runs from -s to +s for a particle of spin s. For a spin-1/2 particle, the eigenvalues are ±ℏ/2.

In a quantum state |ψ⟩, a measurement of the spin along n yields one of the eigenvalues with probability |⟨n, m_s|ψ⟩|^2, where |n, m_s⟩ are the eigenstates of S · n. The eigenstates for different axes form complete bases, and the measurement can be represented by projection operators P_{n, m_s} = |n, m_s⟩⟨n, m_s|. The observable S · n has a spectrum consisting of the eigenvalues m_s ℏ with corresponding eigenvectors |n, m_s⟩.

Mathematically, S · n can be written as S · n = n_x S_x + n_y S_y + n_z S_z. In the spin-1/2 representation, S_x = (ℏ/2) σ_x, S_y = (ℏ/2) σ_y, S_z = (ℏ/2) σ_z, so S · n = (ℏ/2) n · σ, where σ are the Pauli matrices. The eigenvectors of S · n are obtained by rotating the z-axis basis to align with n; the states |n, +⟩ and |n, −⟩ are related to the standard |↑⟩_z and |↓⟩_z by a unitary rotation.

Spin projection is central to experiments such as Stern–Gerlach measurements and to various tasks in quantum information and spin-resolved spectroscopy, where outcomes depend on the projection of spin along chosen directions.