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That is, the resulting spin operators for higher-spin systems in three spatial dimensions can be calculated for arbitrarily large s using this spin operator and ladder operators. For example, taking the Kronecker product of two spin- 1 / 2 yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 ( triplet ...
Operator (physics) 19 languages ... An operator is a function over a space of physical states onto another space of ... (spin, orbital and total) about any two ...
Unlike in more complicated quantum mechanical systems, the spin of a spin- 1 / 2 particle can be expressed as a linear combination of just two eigenstates, or eigenspinors. These are traditionally labeled spin up and spin down. Because of this, the quantum-mechanical spin operators can be represented as simple 2 × 2 matrices.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
To obtain the spinors of physics, such as the Dirac spinor, one extends the construction to obtain a spin structure on 4-dimensional space-time (Minkowski space). Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with SO(3,1) symmetry, and then builds the spin group at each point.
That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3) § A note on Lie algebras. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group ...
All known spin 1 / 2 particles have non-zero mass; however, for hypothetical massless spin 1 / 2 particles (the Weyl spinors), helicity is equivalent to the chirality operator multiplied by 1 / 2 ħ.
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 many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.