Questions tagged [wigner-eckart]

The Wigner–Eckart theorem relates matrix elements of spherical tensor operators in the basis of angular momentum eigenstates to Clebsch–Gordan coefficients. Within a given subspace, a component of such operators behaves proportionally to the same component of the angular momentum operator itself. Do not use for plain Clebsch–Gordan decompositions.

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Wigner-Eckart theorem: Completeness relation

Consider the Wigner-Eckart theorem given by $$\langle \alpha' j m'|A^q|\alpha j m\rangle = \frac{\langle \alpha' j m'|\mathbf{J}\cdot\mathbf{A}|\alpha j m\rangle}{j(j+1)}\langle j m'| J^q|j, m\rangle$$...
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Confusion regarding the Clebsch-Gordan coef. in the Wigner-Eckart theorem

I will start by giving a brief explanation to the Clebsch-Gordan coef. It's because how I perceive this coef. that I don't understand the Wigner-Eckart theorem. The Clebsch-Gordan coefficient related ...
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Product of rank-2 spherical tensors

I have been reading up a lot of material on spherical tensors to try and help me with a calculation of the electric quadrupole energy shift of an atom, and have run into a block due to my lack of ...
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Can we prove this without explicit calculation?

Let $|l,m\rangle$ be standard angular momentum basis. I come across this identity $$\langle2,-1|z|1,-1\rangle=\frac{\sqrt{3}}{2}\langle2,0|z|1,0\rangle$$ Using spherical harmonics, I can see this is ...
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Rotation operator on $\lvert l, m\rangle$ state

Recently I came across Tensor operators and Wigner Ekart theorem , in one its derivations it was given that $\langle l',m'|\mathcal{D}_R |l,m\rangle = \delta_{ll'} D_{mm'}^l(R)$ . Can I get an idea ...
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Do good quantum numbers matter for the Wigner-Eckart theorem?

I have a question related to the following passage in the quantum mechanical scattering textbook by Taylor, Here Taylor makes the choice to use a basis of total angular momentum eigenvectors instead ...
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Proof of Wigner-Eckart Theorem

During my class we've proven the Wigner-Eckart Theorem for irreducible tensor operators. However the proof given to the class by our teacher seems to miss something to actually complete the proof. I ...
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Can this matrix be evaluated with the help of the Wigner-Eckart theorem?

I wonder if the problem in the image can be solved with the Wigner-Eckart (W-E) theorem. These elements have to vanish. I tried introducing the identity operator in between $r$ and $p$ to then use the ...
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