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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 is the angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
 intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 about an external point. In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, spin is particularly important for systems at atomic length scales, such as individual atoms, protons, or electrons. Such particles and the spin of quantum mechanical systems ("particle spin") possesses several unusual or non-classical features, and for such systems, spin angular momentum cannot be associated with rotation but instead refers only to the presence of angular momentum.






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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 is the angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
 intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 about an external point. In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, spin is particularly important for systems at atomic length scales, such as individual atoms, protons, or electrons. Such particles and the spin of quantum mechanical systems ("particle spin") possesses several unusual or non-classical features, and for such systems, spin angular momentum cannot be associated with rotation but instead refers only to the presence of angular momentum. A spin triplet is a set of three quantum states of a system, each with total spin S = 1. The system could consist of a single elementary massive spin 1 particle such as a W or Z boson, or be some multiparticle state with total spin angular momentum of one (in units of ).

Two Spin 1/2 Particles

In a system with two spin 1/2 particles - for example the proton and electron in the ground state of hydrogen, measured on a given axis, each particle can be either spin up or spin down so the system has four basis states in all

using the single particle spins to label the basis states, where the first and second arrow in each combination indicate the spin direction of the first and second particle respectively.

More rigorously

and since for spin-1/2 particles, the basis states span a 2-dimensional space. Therefore the basis states span a 4-dimensional space.

Now the total spin and its projection onto the previously defined axis can be computed using the rules for adding angular momentum in quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 using the Clebsch-Gordan coefficients
Clebsch-Gordan coefficients

In physics, the Clebsch?Gordan coefficients are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics.In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irred...
. In general

substituting in the four basis states


returns the possible values for total spin given along with their representation in the basis. There are three states with total angular momentum 1

and a fourth with total angular momentum 0.

The result is that a combination of two spin 1/2 particles can carry a total spin of 1 or 0, depending on whether they occupy a triplet or singlet state.

See also

  • Singlet state
  • Doublet state
  • Diradical
    Diradical

    A diradical in organic chemistry is a molecule with two electrons occupying two degenerate molecular orbitals . They are known by their higher reactivities and shorter lifetimes....
  • Angular momentum
    Angular momentum

    In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
  • Pauli matrices
    Pauli matrices

    The Pauli matrices are a set of 2 × 2 complex number Hermitian matrix and Unitary matrix matrix Usually indicated by the Greek letter 'sigma' , they are occasionally denoted with a 'tau' when used in connection with isospin symmetries....
  • Spin multiplicity
    Spin multiplicity

    In quantum mechanics, spin multiplicity denotes the number of possible quantum states of a system with given principal spin quantum number S. The different states are distinguished by the spin projection quantum number Sz, which can take the values -S, -S+1, ..., S-1, S....
  • Spin quantum number
    Spin quantum number

    In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given Elementary particle....
  • Spin-1/2
  • Spin tensor
    Spin tensor

    In mathematics and mathematical physics, the Euclidean groupof Euclidean group#Direct and indirect isometries is generated by translations and rotations....
  • Spinor
    Spinor

    In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector and tensor....