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The necessity of introducing half-integer spin goes back experimentally to the results of the Stern–Gerlach experiment.A beam of atoms is run through a strong heterogeneous magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms.
In 1940, Pauli proved the spin–statistics theorem, which states that fermions have half-integer spin, and bosons have integer spin. [7] In retrospect, the first direct experimental evidence of the electron spin was the Stern–Gerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1927. [38]
Fermions have half-integer spin while bosons have integer spin. All the particles of the Standard Model have been experimentally observed, including the Higgs boson in 2012. [ 2 ] [ 3 ] Many other hypothetical elementary particles, such as the graviton , have been proposed, but not observed experimentally.
It is an integer for all bosons, such as photons, and a half-odd-integer for all fermions, such as electrons and protons. The component of the spin along a specified axis is given by the spin magnetic quantum number , conventionally written m s .
This problem is overcome in different ways depending on particle spin–statistics. For a state of integer spin the negative norm states (known as "unphysical polarization") are set to zero, which makes the use of gauge symmetry necessary. For a state of half-integer spin the argument can be circumvented by having fermionic statistics. [21]
Fermions have a half-integer spin (spin 1 / 2 , spin 3 / 2 , etc.) and obey the Pauli exclusion principle. These particles include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.
Half-integer spin means that the intrinsic angular momentum value of fermions is = / (reduced Planck constant) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics, fermions are described by antisymmetric states. In contrast, particles with integer spin (bosons) have symmetric wave functions and may share the same ...
The set of all half-integers is often denoted + = . The integers and half-integers together form a group under the addition operation, which may be denoted [2] . However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. = . [3] The smallest ring containing them is [], the ring of dyadic rationals.