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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.
[34] [6] In 1927, Pauli formalized the theory of spin using the theory of quantum mechanics invented by Erwin Schrödinger and Werner Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators and introduced a two-component spinor wave-function. Pauli's theory of spin was non-relativistic.
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 antihermitian matrices used above are utilized as spin operators, after they are multiplied by i, so they are now hermitian (like the Pauli matrices). Thus, in this language, Thus, in this language,
Pauli matrices, also called the "Pauli spin matrices". Generalizations of Pauli matrices; Gamma matrices, which can be represented in terms of the Pauli matrices.
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 ...
In quantum mechanics, eigenspinors are thought of as basis vectors representing the general spin state of a particle. Strictly speaking, they are not vectors at all, but in fact spinors. For a single spin 1/2 particle, they can be defined as the eigenvectors of the Pauli matrices.
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 =.