In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of 1/2. The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of 1/2 means that the particle must be rotated by two full turns (through 720°) before it has the same configuration as when it started.
Particles having net spin 1/2 include the proton, neutron, electron, neutrino, and quarks. The dynamics of spin-1/2 objects cannot be accurately described using classical physics; they are among the simplest systems which require quantum mechanics to describe them. As such, the study of the behavior of spin-1/2 systems forms a central part of quantum mechanics.
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. It was found that for silver atoms, the beam was split in two—the ground state therefore could not be an integer, because even if the intrinsic angular momentum of the atoms were the smallest (non-zero) integer possible, 1, the beam would be split into 3 parts, corresponding to atoms with Lz = −1, +1, and 0, with 0 simply being the value known to come between −1 and +1 while also being a whole-integer itself, and thus a valid quantized spin number in this case. The existence of this hypothetical "extra step" between the two polarized quantum states would necessitate a third quantum state; a third beam, which is not observed in the experiment. The conclusion was that silver atoms had net intrinsic angular momentum of 1/2.
Spin-1/2 objects are all fermions (a fact explained by the spin–statistics theorem) and satisfy the Pauli exclusion principle. Spin-1/2 particles can have a permanent magnetic moment along the direction of their spin, and this magnetic moment gives rise to electromagnetic interactions that depend on the spin.
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Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin should not be understood as in the "rotating internal mass" sense: spin is a quantized wave property. The existence of electron 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.
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually denoted ) lowers the number of particles in a given state by one. A creation operator (usually denoted ) increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator.
In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron (the others being the principal quantum number n, the magnetic quantum number m_l, and the spin quantum number m_s). It is also known as the orbital angular momentum quantum number, orbital quantum number, subsidiary quantum number, or second quantum number, and is symbolized as l (pronounced ell).
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