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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.
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 ...
For a nonrelativistic spin-1/2 particle of mass m, a representation of the time-independent Lévy-Leblond equation reads: [1] {+ = + =where c is the speed of light, E is the nonrelativistic particle energy, = is the momentum operator, and = (,,) is the vector of Pauli matrices, which is proportional to the spin operator =.
Pauli matrices, also called the "Pauli spin matrices". Generalizations of Pauli matrices; Gamma matrices, which can be represented in terms of the Pauli matrices.
The traditional Pauli matrices are the matrix representation of the () Lie algebra generators , , and in the 2-dimensional irreducible representation of SU(2), corresponding to a spin-1/2 particle. These generate the Lie group SU(2) .
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 two-component helicity eigenstates satisfy ^ (^) = (^) where are the Pauli matrices, ^ is the direction of the fermion momentum, = depending on whether spin is pointing in the same direction as ^ or opposite.
The mysterious underlying property was spin. The first two-dimensional spin matrices (better known as the Pauli matrices) were introduced by Pauli in the Pauli equation; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles in magnetic fields, but this was phenomenological.