Wigner–Eckart theorem

The Wigner- Eckart theorem ( to Eugene Paul Wigner and Carl Henry Eckart ) is a tool for calculating the matrix elements of a tensor operator, if its symmetry properties are known. The Wigner- Eckart theorem should not be confused with the Wigner theorem.

For the defining transformation properties of a tensor operator shall:

With the unitary group transformation matrix and an irreducible representation of this group in the base.

Theorem: The matrix element of a spherical tensor expressed in terms of the eigenstates of the angular momentum operator satisfies the following equation:

Wherein the reduced matrix element (indicated by the two lines on either side of ) is independent of m and m ' and q. It is also preferred, since this of m and m 'are independent matrix element is calculated once and is then the same for all the other matrix elements, thus enabling an easy calculation of any matrix elements.

This is a tensor of rank k, j of the total angular momentum, m is the magnetic quantum number associated and all necessary for the description of the system of the state of quantum numbers.

For rotational symmetry are the Clebsch - Gordan coefficients for the addition of two angular momenta and and the respective z- components or the angular momentum z - component.

Proof of the theorem ( the rotation group )

The Wigner- Eckart theorem is related to the lemma of Schur. When taking advantage of this, longer bills for the proof are not required.

To bring the Clebsch - Gordan coefficients into the game, we consider the following, designed just for this purpose operator:

He transformed states with two angular momenta ( and ) in states with a single angular momentum, which act on tensor. At the finish area rotations are represented by a unitary operator, in the archetype space by a unitary operator. The essential characteristic of the swapping with rotations or the invariance under rotations:

This is due to the similar behavior of tensor and angular momentum states under rotations. Specifically, you can see the invariance of the easiest, by the expression

One evaluates by summation over, which results in, and once by summation over, resulting. This is used in that the rotation matrices are unitary.

Because of the rotational invariance of are subspaces that are irreducible under transformed into subspaces that are irreducible. When the rotation group these subspaces are characterized by an angular momentum quantum number. After Schur's Lemma now applies:

• The parts of that mediate between different ( inequivalent irreducible representations ), are zero.
• The parts of that mediate between the same ( equivalent irreducible representations with the same representation matrices ) are multiples of one mapping.

The fact that the representation matrices for the same angular momenta are actually always the same, based on the use of the standard basis vectors. The proof given here applies only reason for the rotation group.

If the respective multiple is designated by a factor which depends on the associated sub-regions, having on the Schur's Lemma therefore of the form:

The sum over is the one mapping between two irreducible subspaces dar. In Bra- vector of the degeneracy index is missing, because there is no degeneracy in angular momentum coupling ( the archetype of space ). The indices to express that the whole construction of the operator depends on them.

To complete the proof, it now forms the matrix element with the two expressions for, takes advantage of the orthonormality of the basis vectors, and identifies the respective with.