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Classically we have for the angular momentum =. This is the same in quantum mechanics considering and as operators. Classically, an infinitesimal rotation of the vector = (,,) about the -axis to ′ = (′, ′,) leaving unchanged can be expressed by the following infinitesimal translations (using Taylor approximation):
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry ...
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
Dextrorotation and laevorotation (also spelled levorotation) [1] [2] in chemistry and physics are the optical rotation of plane-polarized light.From the point of view of the observer, dextrorotation refers to clockwise or right-handed rotation, and laevorotation refers to counterclockwise or left-handed rotation.
The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the special orthogonal group SO(3). (The '3' refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is SO(3). Any rotation preserves distances on the surface of the ball.
A positive rotation direction usually means one that corresponds to the translation direction by the right-hand rule. This means that if the rotation is clockwise, the displacement is away from the viewer. Except for φ = 180°, we have to distinguish a screw displacement from its mirror image. Unlike for rotations, a righthand and lefthand ...
Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the n th order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of (180°, 120°, 90°, 72°, 60°, 51 3 ⁄ 7 °, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all ...
The rotation also breaks the symmetry in the overall Lorentz transformation making it nonsymmetric. For this specific rotation, let the angle be ε and the axis be defined by the unit vector e, so the axis-angle vector is ε = εe. Altogether, two different orderings of two boosts means there are two inequivalent transformations.