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The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ 1, iσ 2, iσ 3 form a basis for the real Lie algebra (), which exponentiates to the special unitary group SU(2).
Heuristic depiction of spin angular momentum cones for a spin- 1 / 2 particle. Spin- 1 / 2 objects are all fermions (a fact explained by the spin–statistics theorem) and satisfy the Pauli exclusion principle. Spin- 1 / 2 particles can have a permanent magnetic moment along the direction of their spin, and this magnetic ...
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 ...
Atoms can have different overall spin, which determines whether they are fermions or bosons: for example, helium-3 has spin 1/2 and is therefore a fermion, whereas helium-4 has spin 0 and is a boson. [2]: 123–125 The Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability to the chemical behavior ...
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 =.
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).
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.
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 ...