# Wigner-Eckart theorem in Stark effect

• Goddar
In summary, the conversation discusses the use of the Wigner-Eckart theorem in degenerate perturbation theory for a hydrogen atom in a constant electric field. The first two questions are easily solved, but the third question asks about the possibility of observing the Stark effect with x-polarized light. The use of the Wigner-Eckart theorem is questioned due to the presence of the electric field, but it may still be applicable to the [21±1] states. The conversation also includes a discussion of spherical vector operators and their relationship to the Wigner-Eckart theorem.
Goddar
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?

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.

## What is the Wigner-Eckart theorem in Stark effect?

The Wigner-Eckart theorem in Stark effect is a mathematical tool used to simplify the calculation of transition probabilities between different energy levels in a system under the influence of an external electric field.

## How does the Wigner-Eckart theorem relate to Stark effect?

The Wigner-Eckart theorem provides a framework for calculating the matrix elements of the electric dipole moment operator, which is a key factor in determining the Stark effect in a system.

## What is the significance of the Wigner-Eckart theorem in studying Stark effect?

The Wigner-Eckart theorem allows for a more efficient and accurate calculation of the Stark effect, as it reduces the number of integrals that need to be evaluated and provides a systematic way to organize the relevant terms in the calculation.

## Can the Wigner-Eckart theorem be applied to other physical phenomena besides Stark effect?

Yes, the Wigner-Eckart theorem is a general mathematical theorem that can be applied to a wide range of physical systems and phenomena, including atomic and molecular spectroscopy, nuclear physics, and solid state physics.

## Are there any limitations or assumptions associated with the Wigner-Eckart theorem in Stark effect?

Yes, the Wigner-Eckart theorem assumes that the external electric field is weak and that the system can be described by a set of discrete energy levels. It also assumes that the system is in a state of thermal equilibrium.

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