Spin (physics)

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.Working in the coordinate system where θ ^ = z ^ { extstyle {hat { heta }}={hat {z}}} , we would like to show that Sx and Sy are rotated into each other by the angle θ. Starting with Sx. Using units where ħ = 1: In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. In some ways, spin is like a vector quantity; it has a definite magnitude, and it has a 'direction' (but quantization makes this 'direction' different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number. The SI unit of spin is the (N·m·s) or (kg·m2·s−1), just as with classical angular momentum. In practice, spin is given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ, which has the same units of angular momentum, although this is not the full computation of this value. Very often, the 'spin quantum number' is simply called 'spin', leaving its meaning as the unitless 'spin quantum number' to be inferred from context. When combined with the spin-statistics theorem, the spin of electrons results in the Pauli exclusion principle, which in turn underlies the periodic table of chemical elements. Wolfgang Pauli in 1924 was the first to propose a doubling of electron states due to a two-valued non-classical 'hidden rotation'. In 1925, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the old quantum theory of Bohr and Sommerfeld. Ralph Kronig anticipated the Uhlenbeck-Goudsmit model in discussion with Hendrik Kramers several months earlier in Copenhagen, but did not publish. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it. As the name suggests, spin was originally conceived as the rotation of a particle around some axis. This picture is correct so far as spin obeys the same mathematical laws as quantized angular momenta do. On the other hand, spin has some peculiar properties that distinguish it from orbital angular momenta: The conventional definition of the spin quantum number, s, is s = n/2, where n can be any non-negative integer. Hence the allowed values of s are 0, 1/2, 1, 3/2, 2, etc. The value of s for an elementary particle depends only on the type of particle, and cannot be altered in any known way (in contrast to the spin direction described below). The spin angular momentum, S, of any physical system is quantized. The allowed values of S are where h is the Planck constant and ℏ {displaystyle hbar } = h/2π is the reduced Planck constant. In contrast, orbital angular momentum can only take on integer values of s; i.e., even-numbered values of n.