example, the spin projection operator Sz affects a measurement of the spin in the z direction. The two eigenvalues of Sz, , then correspond to the following eigenspinors: These vectors form a complete basis for the Hilbert space describing the spin- particle. Thus, linear combinations of these two states can represent all possible states of the spin, including in the x- and y-directions. The ladder operators are: Since , it follows that and . Thus: Their normalized eigenspinors can be found in the usual way. For Sx, they are: For Sy, they are: RQM (relativistic quantum mechanics) While NRQM defines spin with 2 dimensions in Hilbert space with dynamics that are described in 3-dimensional space and time, relativistic quantum mechanics defines the spin with 4 dimensions in Hilbert space and dynamics described by 4-dimensional space-time. Observables As a consequence of the four-dimensional nature of space-time in relativity, relativistic quantum mechanics uses 4×4 matrices to describe spin operators and observables. Spin as a consequence of combining quantum theory and special relativity When physicist Paul Dirac tried to modify the Schrödinger equation so that it was consistent with Einstein's theory of relativity, he found it was only possible by including matrices in the resulting Dirac equation, implying the wave must have multiple components leading to spin. See also Projective representation Notes Further reading External links Rotation in three dimensions Quantum models