are traditionally labeled spin up and spin down. Because of this, the quantum-mechanical spin operators can be represented as simple 2 × 2 matrices. These matrices are called the Pauli matrices. Creation and annihilation operators can be constructed for spin- objects; these obey the same commutation relations as other angular momentum operators. Connection to the uncertainty principle One consequence of the generalized uncertainty principle is that the spin projection operators (which measure the spin along a given direction like x, y, or z) cannot be measured simultaneously. Physically, this means that the axis about which a particle is spinning is ill-defined. A measurement of the z-component of spin destroys any information about the x- and y-components that might previously have been obtained. Mathematical description A spin- particle is characterized by an angular momentum quantum number for spin s of . In solutions of the Schrödinger equation, angular momentum is quantized according to this number, so that total spin angular momentum However, the observed fine structure when the electron is observed along one axis, such as the z-axis, is quantized in terms of a magnetic quantum number, which can be viewed as a quantization of a vector component of this total angular momentum, which can have only the values of . Note that these values for angular momentum are functions only of the reduced Planck constant (the angular momentum of any photon), with no dependence on mass or charge. Complex phase Mathematically, quantum mechanical spin is not described by a vector as in classical angular momentum. It is described by a complex-valued vector with two components called a spinor. There are subtle differences between the behavior of spinors and vectors under coordinate rotations, stemming from the behavior of a vector space over a complex field. When a spinor is rotated by 360° (one full turn), it transforms to its negative, and then after a further rotation of 360° it transforms back to its initial value again. This is because in quantum theory the state of a particle or system is represented by a complex probability amplitude (wavefunction) ψ, and when the system is measured, the probability of finding the system in the state ψ equals , the absolute square (square of the absolute value) of the amplitude. In mathematical terms, the quantum Hilbert space carries a projective representation of the rotation group SO(3). Suppose a detector that can be rotated measures