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Wigner-Eckart theorem: what is $q $? How does $T^{(k)}_{q}$ relate to an operator?

Physics Asked on July 4, 2021

We saw in class that the Wigner-Eckart theorem is,

$$
langle alpha’, j’,m’|T^{(k)}_{q}|alpha, j, m rangle = langle j,k;m,q|j’,m’rangle frac{langle alpha’, j’||T^{(k)}||alpha, j rangle}{sqrt{2j + 1}}
$$

In the context of the perturbation theroy, if we consider a perturbation of the Hamiltonian,

$$
Delta V = eEz
$$

For the hydrogen atom inside an electric field along $z$, neglecting the spin of the electron, we saw that we could write the spherical tensor as
$$
T^{(k)}_{q} = T^{(1)}_0 = eEz
$$

I was wondering, how can we know that $q$ here is equals to $0$ ? It’s maybe obvious but I’m a little bit lost about this spherical tensor and Wigner-Eckart Theorem.

One Answer

In your notation $k$ is the degree and $q$ the component of the tensor. You can easily verify that $z$ is proportional to the $0$th component since the components satisfy $[L_z,T^{(k)}_q]=q T^{(k)}_q$. Moreover $zsim Y_{10}(theta, phi)$, confirming the value $q=0$ for the component (and for that matter that $k=1$).

Correct answer by ZeroTheHero on July 4, 2021

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