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The term spin matrix refers to a number of matrices, which are related to Spin (physics). Quantum mechanics and pure mathematics.
Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ 0), the Pauli matrices form a basis of the vector space of 2 × 2 Hermitian matrices over the real numbers, under addition.
A normalized spinor for spin- 1 / 2 in the (u x, u y, u z) direction (which works for all spin states except spin down, where it will give 0 / 0 ) is + (+ +). The above spinor is obtained in the usual way by diagonalizing the σ u matrix and finding the eigenstates corresponding to the eigenvalues.
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 the spin angular momentum about for example the -axis we just replace with = (where is the Pauli Y matrix) and we get the spin rotation operator (,) = (). Effect on the spin operator and quantum states
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, [ m ] realizing it as a group of rotations among them, [ n ] but it also acts on the column ...
The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics.The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1).