Vector Operators and the Wigner Eckart Theorem

There are some general features that we can derive about operators which are vectors, that is, operators that transform like a vector under rotations. We have seen in the sections on the Electric Dipole approximation and subsequent calculations that the vector operator $\vec{r}$ could be written as its magnitude $r$ and the spherical harmonics $Y_{1m}$. We found that the $Y_{1m}$ could change the orbital angular momentum (from initial to final state) by zero or one unit. This will be true for any vector operator.

In fact, because the vector operator is very much like adding an additional $\ell=1$ to the initial state angular momentum, Wigner and Eckart proved that all matrix elements of vector operators can be written as a reduced matrix element which does not depend on any of the $m$, and Clebsch-Gordan coefficients. The basic reason for this is that all vectors transform the same way under rotations, so all have the same angular properties, being written in terms of the $Y_{1m}$.

Note that it makes sense to write a vector $\vec{V}$ in terms of the spherical harmonics using

$$\bgroup\color{black} V_\pm=\mp {V_x\pm iV_y\over\sqrt{2}} \egroup$$

and

$$\bgroup\color{black} V_0=V_z. \egroup$$

We have already done this for angular momentum operators.

Lets consider our vector $V^q$ where the integer $q$ runs from -1 to +1. The Wigner-Eckart theorem says

$$\bgroup\color{black} \langle \alpha'j'm'\vert V^q\vert\alpha ... ... \langle \alpha'j'\vert\vert V\vert\vert\alpha j\rangle \egroup$$

Here $\alpha$ represents all the (other) quantum numbers of the state, not the angular momentum quantum numbers. $jm$ represent the usual angular momentum quantum numbers of the states. $\langle \alpha'j'\vert\vert V\vert\vert\alpha j\rangle$ is a reduced matrix element. Its the same for all values of $m$ and $q$. (Its easy to understand that if we take a matrix element of $10r$ it will be 10 times the matrix element of $r$. Nevertheless, all the angular part is the same. This theorem states that all vectors have essentially the same angular behavior. This theorem again allows us to deduce that $\Delta\ell=-1,0.+1$.

The theorem can be generalized for spherical tensors of higher (or even lower) rank than a vector.