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Every rotation in three dimensions is defined by its axis (a vector along this axis is unchanged by the rotation), and its angle — the amount of rotation about that axis (Euler rotation theorem). There are several methods to compute the axis and angle from a rotation matrix (see also axis–angle representation ).
The matrix P corresponding to the position ... (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin 1 ...
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
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 term spin matrix refers to a number of matrices, which are related to Spin (physics). Quantum mechanics and pure mathematics.
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.
An infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation. While a rotation matrix is an orthogonal matrix = representing an element of () (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix = in the tangent space (the special orthogonal Lie ...
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3) , the group of all rotation matrices ...