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Wigner-Eckart theorem in Stark effect

  1. Dec 28, 2014 #1
    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..
  2. jcsd
  3. Dec 29, 2014 #2
    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.
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