In quantum mechanics, electron spin states in three-dimensional space (Z-X-Y coordinates) are defined by linear superpositions where a particle exists simultaneously as probability amplitudes for orthogonal basis vectors until measurement occurs upon which the state collapses to an eigenvalue of +1 or -1. The theory utilizes Hermitian operators known as Pauli matrices ($\Sigma_Z, \Sigma_X, \Sigma_Y$) acting on Hilbert space eigenvectors, where general spin states along arbitrary unit vector directions $\mathbf{n}$ are constructed via the linear combination $n_x\Sigma_X + n_y\Sigma_Y + n_z\Sigma_Z$. This framework establishes that measurement outcomes in any Cartesian direction follow specific probability distributions determined by spherical trigonometric functions (cosine and sine squared of half-angles), fundamentally distinguishing quantum spin from classical rotation due to intrinsic quantization.
In quantum mechanics, electron spin states in three-dimensional space (Z-X-Y coordinates) are defined by linear superpositions where a particle exists simultaneously as probability amplitudes for ort…