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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 ...
The term spin matrix refers to a number of matrices, which are related to Spin (physics). Quantum mechanics and pure mathematics.
Suppose there is a spin 1/2 particle in a state = [].To determine the probability of finding the particle in a spin up state, we simply multiply the state of the particle by the adjoint of the eigenspinor matrix representing spin up, and square the result.
The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple. [7] Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G n is defined to consist of all n-fold tensor products of Pauli matrices.
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 states) and a 1-dimensional spin-0 representation (singlet state). The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis:
These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. [1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark ...
In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, [i] and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of ...
On an eigenspace of the 4-momentum operator with 4-momentum eigenvalue of the Hilbert space of a quantum system (or for that matter the standard representation with ℝ 4 interpreted as momentum space acted on by 5×5 matrices with the upper left 4×4 block an ordinary Lorentz transformation, the last column reserved for translations and the ...
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