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The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
The Pauli matrices are a vector of three 2×2 matrices that are used as spin operators. 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 ...
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).
The term spin matrix refers to a number of matrices, which are related to Spin ... Pauli matrices, also called the "Pauli spin matrices". Generalizations of Pauli ...
The Pauli group is a representation of the gamma group in three-dimensional Euclidean space. It is not isomorphic to the gamma group; it is less free, in that its chiral element is σ 1 σ 2 σ 3 = i I {\displaystyle \sigma _{1}\sigma _{2}\sigma _{3}=iI} whereas there is no such relationship for the gamma group.
Pauli introduced the 2×2 Pauli matrices as a basis of spin operators, thus solving the nonrelativistic theory of spin. This work, including the Pauli equation , is sometimes said to have influenced Paul Dirac in his creation of the Dirac equation for the relativistic electron, though Dirac said that he invented these same matrices himself ...
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.
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 =.