It states that matrix elements of spherical tensor operators in the basis of angular momentum 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, 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 matrix element one finds for the spherical tensor operator is proportional to a Clebsch–Gordan coefficient, which arises when considering adding two angular momenta.
This definition is given in the book Quantum Mechanics by Cohen–Tannoudji, Diu and Laloe.
, where ri is either the x, y, or z component of the position operator, and m1, m2 are the magnetic quantum numbers that distinguish different orbitals within the 2p or 4d subshell.
The Wigner–Eckart theorem allows one to obtain the same information after evaluating just one of those 45 integrals (any of them can be used, as long as it is nonzero).
Then the other 44 integrals can be inferred from that first one—without the need to write down any wavefunctions or evaluate any integrals—with the help of Clebsch–Gordan coefficients, which can be easily looked up in a table or computed by hand or computer.
Finally, an analogous statement is true for the position operator: when the system is rotated, the three different components of the position operator are effectively interchanged or mixed.
This gives an algebraic relation involving K and some or all of the 44 unknown matrix elements.
Different rotations of the system lead to different algebraic relations, and it turns out that there is enough information to figure out all of the matrix elements in this way.
But this is fundamentally the same thing, because of the close mathematical relation between rotations and angular momentum operators.)
To state these observations more precisely and to prove them, it helps to invoke the mathematics of representation theory.
For example, the set of all possible 4d orbitals (i.e., the 5 states m = −2, −1, 0, 1, 2 and their quantum superpositions) form a 5-dimensional abstract vector space.
The Wigner–Eckart theorem works because the direct product decomposition contains one and only one spin-0 subspace, which implies that all the matrix elements are determined by a single scale factor.
is equivalent to calculating the projection of the corresponding abstract vector (in 45-dimensional space) onto the spin-0 subspace.
The key qualitative aspect of the Clebsch–Gordan decomposition that makes the argument work is that in the decomposition of the tensor product of two irreducible representations, each irreducible representation occurs only once.
[4] Starting with the definition of a spherical tensor operator, we have which we use to then calculate If we expand the commutator on the LHS by calculating the action of the J± on the bra and ket, then we get We may combine these two results to get This recursion relation for the matrix elements closely resembles that of the Clebsch–Gordan coefficient.
In fact, both are of the form Σc ab, c xc = 0.
We therefore have two sets of linear homogeneous equations: one for the Clebsch–Gordan coefficients (xc) and one for the matrix elements (yc).
We can only say that the ratios are equal, that is or that xc ∝ yc, where the coefficient of proportionality is independent of the indices.
Hence, by comparing recursion relations, we can identify the Clebsch–Gordan coefficient ⟨j1 m1 j2 (m2 ± 1)|j m⟩ with the matrix element ⟨j′ m′|T(k)q ± 1|j m⟩, then we may write There are different conventions for the reduced matrix elements.
One convention, used by Racah[5] and Wigner,[6] includes an additional phase and normalization factor, where the 2 × 3 array denotes the 3-j symbol.
With this choice of normalization, the reduced matrix element satisfies the relation: where the Hermitian adjoint is defined with the k − q convention.
Although this relation is not affected by the presence or absence of the (−1)2 k phase factor in the definition of the reduced matrix element, it is affected by the phase convention for the Hermitian adjoint.
Another convention for reduced matrix elements is that of Sakurai's Modern Quantum Mechanics: Consider the position expectation value ⟨n j m|x|n j m⟩.
This matrix element is the expectation value of a Cartesian operator in a spherically symmetric hydrogen-atom-eigenstate basis, which is a nontrivial problem.
(In fact, we could obtain the solution quickly using parity, although a slightly longer route will be taken.)
To find the expectation value, we set n′ = n, j′ = j, and m′ = m. The selection rule for m′ and m is m ± 1 = m′ for the T(1)±1 spherical tensors.
As we have m′ = m, this makes the Clebsch–Gordan Coefficients zero, leading to the expectation value to be equal to zero.