Rotational invariance

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a function

Function (mathematics)

In mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...

 defined on an inner product space

Inner product space

In mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

 is said to have rotational invariance if its value does not change when arbitrary rotation

Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...

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 operator

Operator

In mathematics, an operator is a type of function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the...

 that acts on such functions, rotational invariance may also mean that the function or operator commutes with rotations of X.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a function

Function (mathematics)

In mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...

 defined on an inner product space

Inner product space

In mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

 is said to have rotational invariance if its value does not change when arbitrary rotation

Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...

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 operator

Operator

In mathematics, an operator is a type of function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the...

 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

Laplace operator

In mathematics and physics, the Laplace operator or Laplacian, named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. It is denoted by the symbols Δ, ∇², or ∇·∇. In physics, it is used in the...

 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)); that is, rotating a function merely rotates its Laplacian.

Application to quantum mechanics

In quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...

, rotational invariance is the property that after a rotation

Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...

 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 dθ 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 momentum

Angular momentum

Angular momentum is a quantity that is useful in describing the rotational state of a physical system. For a rigid body rotating around an axis of symmetry , the angular momentum can be expressed as the product of the body's moment of inertia and its angular velocity...

is conserved.