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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. [1] [2]: 183–184 Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory.
In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).
In quantum mechanics, a quantum state is typically represented as an element of a complex Hilbert space, for example, the infinite-dimensional vector space of all possible wavefunctions (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically.
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 component of the spin along a specified axis is given by the spin magnetic quantum number, conventionally written m s. [ 1 ] [ 2 ] The value of m s is the component of spin angular momentum, in units of the reduced Planck constant ħ , parallel to a given direction (conventionally labelled the z –axis).
In quantum mechanics, a triplet state, or spin triplet, is the quantum state of an object such as an electron, atom, or molecule, having a quantum spin S = 1. It has three allowed values of the spin's projection along a given axis m S = −1, 0, or +1, giving the name "triplet".
In quantum field theory, in the case of a massive field, the Casimir invariant W μ W μ describes the total spin of the particle, with eigenvalues = = (+), where s is the spin quantum number of the particle and m is its rest mass.
The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group [26] (the half-spin representations, or Weyl spinors) via + =, =. When dim( V ) is odd, V = W ⊕ U ⊕ W ′ , where U is spanned by a unit vector u orthogonal to W .