In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. For example, the function f(x,y) = x² + y² is invariant under rotations of the plane around the origin. Euclid, detail from The School of Athens by Raphael. ... Partial plot of a function f. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called scalar product or dot product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ...

For a function from a space X to itself, or for an operator 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 operator Δ f = ∂_xx f + ∂_yy f:  if g is the function g(p) = f(r(p)), where r is any rotation, then (Δ g)(p) = (Δ f)(r(p)) -- i.e., rotating a function merely rotates its Laplacian. In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... Mathematical meaning In mathematics, especially abstract algebra, a binary operation on a set S is commutative if for all x and y in S. Otherwise, the operation is noncommutative. ... In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications in mathematics and physics. ...

See also isotropic, Maxwell's theorem, rotational symmetry.

In quantum mechanics, rotational invariance is the property that after a rotation the new system still follows Schrödinger's equation. That is Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... In probability theory, Maxwells theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable X = ( X1, ..., Xn )T is the same as the distribution of GX for every n×n orthogonal matrix G and the components are independent, then... Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...

[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 dθ the rotation operator is In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ...

R = 1 + J_z dθ,

[1 + J_z dθ, d/dt] = 0;

thus

d/dt(J_z) = 0,

in other words angular momentum is conserved. In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ...

See also

• Isotropy