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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.

The Wigner–Eckart theorem relates matrix elements of spherical tensor operators in the basis of angular momentum eigenstates to Clebsch–Gordan coefficients. Operating with a spherical tensor operator on an angular momentum eigenstate is then like adding a state with angular momentum to the state: Within a given subspace, a component of this operator behaves proportionally to the same component of the angular momentum operator itself.

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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 it is useful and perhaps just help me understand…
Cogitator
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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, T_q^{\;\;(k)}] = \hbar \left( (k \mp q)(k \pm q…
Friedrich
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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 W-E theorem for the new matrix elements. I was…
Toorple
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Doubt on Sakurai's proof of Wigner-Eckart theorem

In Sakurai's and Napolitano's book "Modern quantum mechanics" there's a nice proof of the theorem. This can be found also almost identical on Wikipedia's Wigner–Eckart theorem - Proof. The thing that is bugging me about this proof is this part We…
pp.ch.te
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Why is a nucleus with spin 1/2 spherical?

So i am trying to understand the relaxation mechanism in NMR for nuclei with spin $I>\frac{1}{2}$. I know that nuclei with $I=\frac{1}{2}$ have spherical shape, because many paper have told me so. I guess the argument is that we calculate the…
Luca
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What is the relationship between "spectroscopic" and "intrinsic" nuclear quadrupole moments?

Several Physics SE questions/answers note the difference between intrinsic and spectroscopic electric quadrupole moments: Why do spin-1/2 nuclei have zero electric quadrupole moment? Measurement of (intrinsic) electric quadrupole moment of…
David Bailey
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Sums of $6j$ symbols -- reference request

In the book Quantum Theory of Angular Momentum by Varshalovich, Moskalev and Khersonskii, page 305 lists many formulas for sums involving Wigner $6j$ symbols, such as $$\sum\limits_X (-1)^{p+q+X}(2X+1) \left\{\begin{matrix} a & b & X \\ c & d &…
Lee
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Electric quadrupole transition matrix element for hyperfine Zeeman levels

I am trying to calculate the electric quadrupole transition matrix elements for an alkali for transitions between hyperfine Zeeman levels. The electric quadrupole Hamiltonian consists of terms of the form $r^2 Y_{2,q}$. I decomposed the hyperfine…
Calm Cookie
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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, the author defines irreducible tensor operators as…
Dzhou
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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 understand some things, when we consider tensor operators…
imbAF
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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 indeed correct, but I am wondering can we deduce…
Apocalypse
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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 of using the simple tensor product given in the…
Jasper
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Wigner-Eckart Theorem

The Wigner-Eckart Theorem states that in state $| j = 1 , m \rangle$ : $$ \langle 1 ,m_1| Y^{m_{2}}_1 |1,m_{2}\rangle = \int d\Omega Y^{{*}^{m_1}}_1 Y^{m_2}_{1} Y^{m_3}_{1} = C^{1 1 1}_{m_{1} m_{2} m_{3}} \langle j|| Y^{m_{2}}_1||j\rangle $$ In the…
RKerr
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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 be found with the Clebsch-Gordon…
dsm
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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 $\mathrm{SU}(2)$. Let $G \subset \mathrm{SU}(2)$ be a finite group…
Eric Kubischta
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