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Rotational invariance

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	Topics Rotational invariance Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , a functionFunction (mathematics) Overview In mathematics, a function relates each of its inputs to exactly one output.... defined on an inner product spaceInner product space Overview In mathematics, an inner product space is a vector space with additional structure, an inner product , which allows us... is said to have rotational invariance if its value does not change when arbitrary rotationRotation Rotation is the movement of an object in a circular motion.... s are applied to its argument. For example, the function f(x,y) = x² + y² is invariant under rotations of the plane around the origin. For a function from a space X to itself, or for an operatorOperator In mathematics, an operator is a function, usually of a special kind depending on the topic.... that acts on such functions, rotational invariance may also mean that the function or operator commutes with rotations of X. An example is the two-dimensional Laplace operatorLaplace operator In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, named after Pierre-Simon Laplace, i... ? f = ?_xx f + ?_yy f: if g is the function g(p) = f(r(p)), where r is any rotation, then (? g)(p) = (? f)(r(p)) -- i.e., rotating a function merely rotates its Laplacian. See also isotropic, Maxwell's theoremMaxwell's theorem In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution... , rotational symmetryRotational symmetry Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space.... . Application to quantum mechanics In quantum mechanicsQuantum mechanics Quantum mechanics is a first quantized quantum theory that supersedes classical mechanics at the atomic and subatomic levels... , rotational invariance is the property that after a rotationRotation Rotation is the movement of an object in a circular motion.... the new system still obeys Schrödinger's equation. That is [R, E − H] = 0 for any rotation R. Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have [R, H] = 0. Since [R, E − H] = 0, and because for infinitesimal rotations (in the xy-plane for this example; it may be done likewise for any plane) by an angle d? the rotation operator is R = 1 + J_z dθ, [1 + J_z dθ, d/dt] = 0; thus d/dt(J_z) = 0, in other words angular momentumAngular momentum Overview In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the... is conserved. See also IsotropyIsotropy Isotropy is the property of being independent of direction....