Rotational invariance

Encyclopedia

In mathematics

, a function

defined on an inner product space

is said to have

s are applied to its argument. For example, the function

For a function from a space

Δ

In physics

, if a system behaves the same regardless of how it is oriented in space, then its Lagrangian

is rotationally invariant. According to Noether's theorem

, if the action

(the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.

,

the new system still obeys Schrödinger's equation. That is

Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have [

Since [

thus

in other words angular momentum

is conserved.

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

defined on an inner product space

Inner product space

In mathematics, an inner product space is a vector space with an 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 rotationRotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s are applied to its argument. For example, the function

*f*(*x*,*y*) =*x*^{2}+*y*^{2}is invariant under rotations of the plane around the origin.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 operatorLaplace operator

In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

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.In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, if a system behaves the same regardless of how it is oriented in space, then its Lagrangian

Lagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

is rotationally invariant. According to Noether's theorem

Noether's theorem

Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...

, if the action

Action (physics)

In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...

(the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.

## Application to quantum mechanics

In quantum mechanicsQuantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

,

**rotational invariance**is the property that after a rotationRotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

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/d*t*] = 0;

thus

- d/d
*t*(*J*_{z}) = 0,

in other words angular momentum

Angular momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

is conserved.