Wigner-Eckart theorem - AbsoluteAstronomy.com

The Wigner–Eckart theorem is a theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

of representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

and quantum mechanics

Quantum 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...

. It states that matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

elements of spherical tensor operator

Operator (physics)

In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

s on the basis of 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...

eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch-Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart

Carl Eckart

Carl Henry Eckart was an American physicist, physical oceanographer, geophysicist, and administrator. He co-developed the Wigner-Eckart theorem and is also known for the Eckart conditions in quantum mechanics.-Education:Eckart began college in 1919 at Washington University in St...

who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.

The Wigner-Eckart theorem reads:

where

is a rank

spherical tensor,

and

are eigenkets of total angular momentum

and its z-component

has a value which is independent of

and

, and

is the Clebsch-Gordan coefficient for adding

and

to get

.

In effect, the Wigner–Eckart theorem says that operating with a spherical tensor operator of rank

on an angular momentum eigenstate is like adding a state with angular momentum

to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch-Gordan coefficient, which arises when considering adding two angular momenta.

Example

Consider the position expectation value

. This matrix element is the expectation value of a Cartesian operator in a spherically-symmetric hydrogen-atom-eigenstate basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

, which is a nontrivial problem. However, using the Wigner–Eckart theorem simplifies the problem. (In fact, we could get the solution right away using parity

Parity (physics)

In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

, but we'll go a slightly longer way.)

We know that