# Wigner-Eckart theorem in Stark effect

Hi. i'm reviewing some past qualifying exams and stumbled on something i can't figure out, probably because i'm still confused about the Wigner-Eckart theorem..
So, the set-up is just degenerate perturbation theory for constant electric field along z on the n = 2 hydrogen states. That's a classic, so i don't have a problem with it: states [200] and [210] emerge in linear combinations, with energy levels split at ±3u (u = Bohr radius times eE) from the n = 2 level. [21±1] are unaffected.

Penultimate question is: "Assuming that the atom is in its ground state and the light is polarized in z- direction, determine frequencies of spectral lines in the absorption spectrum, which will be observed, and their relative strength"
That doesn't seem hard either: jumps from the ground state imply absorption of ΔE or ΔE±3u, where ΔE is the energy difference between ground state and n = 2; frequency is given by E = hf. Then ΔE absorption should be twice the intensity of any of the two others.

Finally the one i'm stuck on: "Can Stark effect be observed with x-polarized light? Give arguments based on Wigner- Eckart theorem"
First of all, what's the deal about x-polarized here and why does it make a difference?
Then, it seems like the electric field ruins the spherical symmetry so how can we use Wigner-Eckart at all?
Maybe just on the [21±1] states?
If we can use it, then how?...Is it the selection rules that prevent [21±1] from being affected?
Thank you in advance..

## Answers and Replies

Maybe this can help: a family of spherical vector operators with ##\ell = 1##
$$\mathbf{v}_{+1}=-\frac{1}{\sqrt{2}}(\mathbf{x} + i \mathbf{y})\,\quad \mathbf{v}_{-1}=\frac{1}{\sqrt{2}}(\mathbf{x} - i \mathbf{y})\,,\quad \mathbf{v}_0 = \mathbf{z}$$

You might be able to use something like
$$\mathbf{x} = \frac{1}{\sqrt{2}} \left( -\mathbf{v}_{+1} + \mathbf{v}_{-1}\right) \;,$$

since ##\mathbf{v}_{+1}## and ##\mathbf{v}_{+1}## can be used with the W-E theorem.