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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Dropped sign from Wigner-Eckart Theorem for matrix elements of $x$ in hydrogen atom $n = 2$ shell

Consider the $n=2$ states of the hydrogen atom, which we label by $|n\,l\,m_l\rangle$. I want to calculate whether or not there should be a sign difference in these specific matrix elements of $x$: $$\...
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Why can an electric dipole transition allow $\Delta L=0,\pm2$ in hydrogen-like atoms?

I am studying the transition rule of electric dipole transition. I have some question about some rules in the Wikipedia: https://en.wikipedia.org/wiki/Selection_rule#Summary_table In that table, they ...
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How to reduce matrix element of a spherical tensor operator?

I am studying about Wigner-Eckart theorem, and I have a question about the reduced matrix element. Wigner-Eckart theorem: (I follow the terms as in Wikipedia, https://en.wikipedia.org/wiki/Wigner%E2%...
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The proof of the Wigner-Eckart theorem for irreducible tensor operators

I am reading through Wu-Ki Tung's Group Theory in Physics and I met a problem when going through the part of the Wigner-Eckart theorem for irreducible tensor operators. In the 4.3 part of the book, ...
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Equivariance and Wigner $D$ matrices for Spherical harmonic rotations

I am trying to understand equivariance in machine learning, specially as discussed in the following paper. Claim is that equivariance is when Group symmetry operation, such as rotation, commutes with ...
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Confusion about the Wigner-Eckart theorem

Background This will be a lengthy thread, but I made sure that all 3 questions are related to each other and related to the same topic. I currently encountered the W.E-theorem and while I do ...
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Sum of squared Wigner $3j$ symbols

Since Wigner $3j$-symbols for $m_1 = m_2 = m_3= 0 $ have the property $$ \sum_{l_3 = |l_1 -l_2|}^{l_1 + l_2}(2 l_3 + 1)\begin{pmatrix} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \\ \end{pmatrix}^2 =...
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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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Wigner-Eckart theorem: what is $q $? How does $T^{(k)}_{q}$ relate to an operator?

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}{\...
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Wigner-Eckart Theorem

The Wigner-Eckart Theorem states that in state $| j = 1 , m \rangle$ : $$ \langle 1 ,m_1| Y^{m_{2}}_1 |1,m_{2}\rangle = \int d\Omega Y^{{*}^{m_1}}_1 Y^{m_2}_{1} Y^{m_3}_{1} = C^{1 1 1}_{m_{1} m_{2} m_{...
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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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Component of a irreducible tensor product

Given the operators $\boldsymbol{\alpha}$ and $\boldsymbol{C^{(L)}}$ such that $$ \boldsymbol{\alpha}=\left(\begin{array}{cc} 0 & \boldsymbol{\sigma}_{p} \\ \boldsymbol{\sigma}_{p} & 0 \end{...
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Computing matrix elements using Wigner-Eckart Theorem

I am trying to compute the matrix elements \begin{equation} \langle j \, m | a_{1}^{+} | j' \, m'\rangle \end{equation} \begin{equation} \langle j \, m | a_{2}^{+} | j' \, m'\rangle \end{equation} ...
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Selection rules with the Wigner-Eckart Theorem

Working in the $|\alpha, j,m_j\rangle$ basis (denoting all irrelevant quantum numbers by $\alpha$), the Wigner-Eckart theorem tells us that the elements of a rank $k$ spherical tensor $T_q^{(k)}$ can ...
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Clebsch-Gordan Coefficient Recursion Derivation

I've got a question regarding a very particular recursion relation that Condon and Shortley use in their book "The Theory of Atomic Spectra," (TAS) which Racah then uses to derive the algebraic form ...
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Are there irreducible tensors of half integral degree in quantum mechanics?

According to Ballentine, an irreducible tensor of degree k can be defined as a set of $2k + 1$ operators $\{T_q^{\;\;(k)}:(-k \le q \le k)\}$ satisfying the following commutation relations: $$ [J_\pm,...
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What is the usefulness of the Wigner-Eckart theorem?

I am doing some self-study in between undergrad and grad school and I came across the beastly Wigner-Eckart theorem in Sakurai's Modern Quantum Mechanics. I was wondering if someone could tell me why ...
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