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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How to go from a vector operator to its components?
(I'm sorry if this question is a duplicate, I couldn't find anything that answered my question.)
I'm doing an exercise where I'm supposed to get the matrix elements for the vector operator $D$ (the ...
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Wigner-Eckart theorem in classical physics?
The Wigner-Eckart theorem is a useful result in quantum physics and its many applications. Most presentations of this material in books on QM and online lecture notes seem to be variations on the same ...
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Wigner-Eckart for Finite groups
We know the Wigner-Eckart theorem generalizes to say $\mathrm{SU}(3)$ (see for example this answer). In a different direction, I am curious if/how it generalizes to finite groups of $\mathrm{SU}(2)$.
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Wigner-Eckart theorem for exponentiated vector operator?
Consider the Hamiltonian of two spinning particles in a magnetic field with $$H = \vec{B}\cdot\vec{\mu}$$ where $$\vec{\mu} = \alpha \vec{L}_1+\beta\vec{L}_2$$
Now I wish to compute its partition ...
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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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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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