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Relativistic spin explained gyromagnetic anomaly. 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.
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.
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]
Fermi–Dirac statistics applies to identical and indistinguishable particles with half-integer spin (1/2, 3/2, etc.), called fermions, in thermodynamic equilibrium. For the case of negligible interaction between particles, the system can be described in terms of single-particle energy states. A result is the Fermi–Dirac distribution of ...
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 ...
Even-even nuclei with even numbers of both protons and neutrons, such as 12 C and 16 O, have spin zero. Odd mass number nuclei have half-integer spins, such as 3 / 2 for 7 Li, 1 / 2 for 13 C and 5 / 2 for 17 O, usually corresponding to the angular momentum of the last nucleon added.
The spin-statistics theorem relates the exchange symmetry of identical particles to their spin. It states that bosons have integer spin, and fermions have half-integer spin. It states that bosons have integer spin, and fermions have half-integer spin.
Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties. First consider the simpler bosonic case of the photons of the quantum harmonic oscillator.