Wigner-Eckart theorem

From Wikipedia, the free encyclopedia

The Wigner-Eckart theorem is a theorem of representation theory and quantum mechanics. It says that expectation values in hydrogen eigenstates of spherical tensor operators involve the use of Clebsch-Gordan coefficients.

The Wigner-Eckart Theorem states

$\langle njm|T^k_q|n'j'm'\rangle =\langle nj||T^k||n'j'\rangle C^{jm}_{kqj'm'}$

where $T^k_q$ is a rank $k$ spherical tensor, $|njm\rangle$ and $|n'j'm'\rangle$ are eigenkets of total angular momentum $J 2$ and its z-component $J z$ , $\langle nj||T^k||n'j'\rangle$ has a value which is independent of $m$ and $q$ , and $C^{jm}_{kqj'm'}$ is a Clebsch-Gordan coefficient.

In effect, the Wigner-Eckart theorem says that operating with a spherical tensor operator of rank k on a hydrogen eigenstate is like adding a state with angular momentum k to the state. The expectation value one finds for the spherical tensor operator is proportional to a Clebsch-Gordan coefficient, which arose when considering adding two angular momenta.

[edit] Example

Consider the position expectation value $\langle njm|x|njm\rangle$ . This matrix element is the expectation value of a Cartesian operator in a spherically-symmetric hydrogen-atom-eigenstate basis, 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, but we'll go a slightly longer way.)

We know that $x$ is one component of $\vec r$ , which is a vector. Vectors are rank-1 tensors, so $x$ is some linear combination of $T^1_q$ for $q = - 1,0,1$ . In fact, it can be shown that $x=\frac{T_{-1}^{1}-T^1_1}{\sqrt{2}}$ . Therefore