Editing Wigner–Eckart theorem (section)

{{short description|Theorem used in quantum mechanics for angular momentum calculations}}
The '''Wigner–Eckart theorem''' is a [[theorem]] of [[representation theory]] and [[quantum mechanics]]. It states that [[Matrix (mathematics)|matrix]] elements of [[spherical tensor operator]]s in the basis of [[angular momentum]] [[eigenstate]]s 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]], 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.<ref name="Eckart Biography">[http://orsted.nap.edu/openbook.php?record_id=571&page=194 Eckart Biography] – The National Academies Press.</ref>

Mathematically, the Wigner–Eckart theorem is generally stated in the following way.  Given a tensor operator <math>T^{(k)}</math> and two states of angular momenta <math>j</math> and <math>j'</math>, there exists a constant <math>\langle j \| T^{(k)} \| j' \rangle</math> such that for all <math>m</math>, <math>m'</math>, and <math>q</math>, the following equation is satisfied:

:<math>
    \langle j \, m | T^{(k)}_q | j' \, m'\rangle
  = \langle j' \, m' \, k \, q | j \, m \rangle \langle j \| T^{(k)} \| j'\rangle,
</math>

where
*<math>T^{(k)}_q</math> is the {{math|''q''}}-th component of the spherical tensor operator <math>T^{(k)}</math> of rank {{math|''k''}},<ref>The parenthesized superscript {{math|(''k'')}} provides a reminder of its rank.  However, unlike {{math|''q''}}, it need not be an actual index.</ref> 
* <math>|j m\rangle</math> denotes an eigenstate of total angular momentum {{math|''J''<sup>2</sup>}} and its ''z'' component {{math|''J''<sub>z</sub>}},
* <math>\langle j' m' k q | j m\rangle</math> is the [[Clebsch–Gordan coefficient]] for coupling {{math|''j''′}} with {{math|''k''}} to get {{math|''j''}},
* <math>\langle j \| T^{(k)} \| j' \rangle</math> denotes<ref>This is a special notation specific to the Wigner–Eckart theorem.</ref> some value that does not depend on {{math|''m''}}, {{math|''m''′}}, nor {{math|''q''}} and is referred to as the '''reduced matrix element'''.

The Wigner–Eckart theorem states indeed that operating with a spherical tensor operator of rank {{math|''k''}} on an angular momentum eigenstate is like adding a state with angular momentum ''k'' 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. When stated another way, one can say that the Wigner–Eckart theorem is a theorem that tells how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator.  This definition is given in the book ''Quantum Mechanics'' by Cohen–Tannoudji, Diu and Laloe.