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1answer
42 views

Angular momentum operator transformation under rotation

The generators of rotations $J_i$ under rotation transform as $$J_i'=R_{ij}J_j\,.$$ Now $J^2=J_1J_1+J_2J_2+J_3J_3$ trasform as $$ \begin{aligned} J'^2&=RJ^2R^{-1}=RJ_1R^{-1}RJ_1R^{-1}+RJ_2R^{-1}...
0
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0answers
33 views

What's the relation between the Lorenztz group and spin of particles?

I know that particles are defined in terms of irreducible representations of the Poincaré group, and that the state of a massive particle is defined by its mass and spin, which are the eigenvalues of ...
2
votes
1answer
36 views

Wicks contractions of stress-energy tensor and plane partitions

I am working out the number of wick contraction of a number $n$ of stress-energy tensor in 4D CFT. The strategy is as follows: For 1 stress energy tensor $T_{\alpha\beta}$, you have only one ...
1
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1answer
52 views

Some Clebsch-Gordan coefficients for $j_{1}=1$ and $j_{2}=1$

I've successfully derived every coefficient, but not the one that has $j=0$. Starting from $|J=2,M=2⟩$ and applying $J_{-}$ we derive $|2,1⟩$ and $|2,0⟩$ and using orthonormality (and the Condon-...
2
votes
1answer
59 views

Symmetries of Wigner $3j$-symbols by exchange

I know that Wigner $3j$-symbols have certain symmetry factors arising by exchange of two columns within one symbol. But what happens if you have two 3j symbols and do an exchange like this: $ \left(\...
0
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0answers
30 views

Symmetry relation of Wigner-Eckart

I saw a symmetry relation following from the the Wigner-Eckart Theorem looking like this $$(\xi j|| T_L || \xi'j') = (-1)^{j-j'} (\xi' j'|| T_L || \xi j)^*$$ I know that it must come somehow under ...
1
vote
1answer
65 views

Representations of the rotation group

(I have already done a similar question, but I did not express myself very well and the question was a bit confusing, so let me try again. If you find the question repetitive, I apologize and you can ...
0
votes
1answer
90 views

$\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
0
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1answer
39 views

Basis function of $\Gamma_2$ irrep of point group $T_d$?

From Properties of the Thirty-Two Point Groups (Koster, et. al.), the basis function of the $\Gamma_2$ irrep of the point group $T_d$ is $l_xl_yl_z$, where $l$ is the angular momentum operator. ...
2
votes
2answers
45 views

Different global phase shifts of Pauli-$z$ Matrix eigenstates from rotations around $z$-axis

I understand the pauli matrix $\sigma_z = \bigl( \begin{smallmatrix}1 & 0\\ 0 & -1\end{smallmatrix}\bigr)$ rotates a state around $z$-axis by angle $\pi$ in $SO(3)$. We can see it works by ...
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2answers
49 views

$j=\frac{1}{2}$ addition of angular momentum

For $j=\frac{1}{2}, j'=\frac{1}{2}$ we have $$|11\rangle=|\frac{1}{2}\frac{1}{2}\rangle$$ $$|10\rangle=\frac{1}{\sqrt{2}}(|-\frac{1}{2}\frac{1}{2}\rangle+ |\frac{1}{2}-\frac{1}{2}\rangle)$$ $$|10\...
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votes
2answers
126 views

Clebsch-Gordan coefficients for more than 2 particles

I need to couple arbitrary spins together and need Clebsch-Gordan coefficients for them. This should be just coupling the last two particles, then couple the next until the first particle is coupled. ...
4
votes
1answer
242 views

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 ...
2
votes
1answer
83 views

Relation Between Cross Product and Infinitesimal Rotations, Generators, Etc [duplicate]

Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group $SO(3)$. For example: $$\vec{\mathbf{...
0
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0answers
86 views

Why are systems joined via a tensor product? [duplicate]

This question comes from seeing that the triangle addition rule for quantum mechanics comes out of groups/representation theory; I thought this was odd as we haven't used any group ideas in QM up to ...
-2
votes
1answer
209 views

Is spin 1 described by $SO(3)$ or $SU(2)$ [duplicate]

What spin is described by which rotation group? I always only find information about spin-1/2
5
votes
2answers
105 views

Operational definition of rotation of particle

The question in brief: what does it mean, operationally, to rotate an electron? Elaboration/background: I am trying to understand how representation theory applies to quantum mechanics. A stumbling ...
1
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1answer
87 views

Does dimension of irreducible representations of the double cover $SU(2)$ of the 3D rotation group define spin of particle?

In quantum field theory, does dimension of irreducible representations of the double cover $SU(2)$ of the 3D rotation group conclusively define spin? In other words, Is spin 1 particle only thing ...
1
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2answers
82 views

Clebsch-Gordan coefficient for 1x0

I'm trying to work out the combination of $|1\ 0 \rangle|0\ 0 \rangle$ (in this case they represent isospin, $|I\ I_3 \rangle$) using Clebsch-Gordan coefficients, but the table for $j_1\times j_2=1\...
0
votes
1answer
93 views

What is a good basis for this Hamiltonian with reduced symmetry?

What would be a good basis for a modified Hamiltonian that reads: $$ H_1 = \frac{1}{2}(L_+S_- + L_-S_+) + c_1 L_x + c_2 S_x,$$ from a symmtry point of view? The Hamiltonian itself is not difficult ...
1
vote
1answer
194 views

Relation between Spin 1 representation and angular momentum and $SO(3)$

This is a naive question. It occurred to me while studying in detail the the Spin 1 angular momentum matrices. The generators of $SO(3)$ are $J_x= \begin{pmatrix} 0&0&0 \\ 0&0&-1 \\ ...
1
vote
1answer
168 views

Symmetric tensor product decomposition of $su(2)$

Taking the tensor product of two spin 1 representations of $su(2)$ yields $$1 \otimes 1 = 0 \oplus 1 \oplus 2.$$ What changes if instead we take the symmetric tensor product $1 \odot 1$ of these ...
0
votes
1answer
265 views

Why we need $SU(2)$ symmetry? When we use it? [closed]

I am trying to learn Quantum mechanics and I am familiar with Pauli matrice but not with group theory. I want to understand SU2 symmetry in common language. When we talk about Pauli matrix x we simply ...
1
vote
1answer
98 views

$SO(3)$, orbital angular momentum, vector product

I have a big confusion with group theory terminology. I know that orbital angular momentum (OAM) is $\mathrm{SO}(3)$-symmetric in 3D-space. Let's define QM orbital angular momentum (OAM) ...
1
vote
1answer
117 views

Rotational invariance of many particle (quantum system) system?

I am trying to prove pairwise coupled harmonically interacting (quantum) system of particles as rotationally invariant . $$H=\frac{1}{2}\sum_{j}p_{j}^2 +\frac{1}{2}k\sum_{i<k}(\vec{r_{i}}-\vec{r_{k}...
1
vote
1answer
321 views

Computing reduced matrix element without Wigner-Eckart theorem

Lets have a problem: suppose we need to calculate reduced matrix element of some transition of a particle from some higher-order spin(or rather total angular momentum state, it does not really matter, ...
2
votes
4answers
451 views

Deriving the unitary operator $U(R)$ associated with a rotation $R$ using Wigner's theorem

A rotation $R(\hat{\textbf{n}},\phi)$ about an arbitrary axis $\hat{\textbf{n}}$ through an angle $\phi$ in the three-dimensional physical space is given by $$R(\hat{\textbf{n}},\phi)=e^{-i(\textbf{j}\...
2
votes
3answers
131 views

What do the antisymmetric matrices $J_i$ represent in classical mechanics?

In physical three-dimensional space, a rotation about an arbitrary axies $\hat{\textbf{n}}$ through an angle $\phi$ can be represented by $$R(\hat{\textbf{n}},\phi)=e^{-i(\textbf{J}\cdot\hat{\textbf{n}...
0
votes
1answer
227 views

If momentum is the generator of translations in position space, does position act similarly in momentum space?

Since $\exp(-ia\hat{p})|x\rangle=|x+a\rangle$, do we find that $\exp(-ik\hat{x})|p\rangle=|p+k\rangle$? If so, I don't think I've seen this symmetry explicitly discussed yet, how do we interpret it? ...
9
votes
2answers
943 views

Why are the generators of rotation in the 4-dimensional Euclidean space correspond to rotations in a plane?

In three-dimensions, the rotation generators are represented by $J_1$, $J_2$ and $J_3$ where $1,2,3$ respectively stands for the generator of rotation about $x,y,z$ axes respectively. In general, in ...
1
vote
1answer
227 views

Connection between spin and lorentz invariance [duplicate]

I came across this statement in the book "Quantum Field theory and the Standard Model" by Matthew Schwartz. "There is a deep connection between Spin and Lorentz invariance that is obscure in Non ...
3
votes
1answer
280 views

Helicity quantization of massless particles

In Appendix B of QFT in a nutshell by Zee, a review of group theory is given. In the last paragraph of the appendix on page 533, he briefly discusses the helicity quantization of massless particles. ...
0
votes
0answers
187 views

6j-symbol from summing Clebsch-Gordan coefficients

I'm trying to do some three body calculations using a harmonic oscillator basis in two relative coordinates. As part of the calculations, I get a sum of a product of multiple Clebsch-Gordan ...
3
votes
1answer
237 views

Quantization of angular momentum in $SO(3)$

When hermitian operators $L_1, L_2, L_3$ follow the commutation relations: $$ [L_1,L_2]=i\;L_3 \\ [L_2,L_3]=i\;L_1 \\ [L_3,L_1]=i\;L_2 $$ one can show that, assuming they are in finite number, their ...
4
votes
2answers
2k views

Physical Interpretation of Clebsch-Gordan Coefficient

I have seen in the algebra of Angular Momentum, how Clebsch-Gordan coefficients arise. However, I do not understand the physical interpretation of it. When I checked the literature it was all in ...
0
votes
1answer
129 views

Matrix of any special unitary transformation in two dimensions

I want to show that every special unitary transformation in two dimensions can be written as the matrix $$ U = \left(\begin{array}{cc} e^{i(\delta + \varphi)}\cos\theta & i~e^{i(\delta - \varphi)}...
1
vote
1answer
274 views

Queries of Proof of Wigner-Eckart Theorem

With regard to the Wigner-Eckart Theorem the following is stated: The following is an outline of the proof in a text I am using: "Consider the action of a tensor-operator component on an angular-...
5
votes
6answers
548 views

Queries about rotational groups $\mathrm{SO}(3)$ and $\mathrm{SU}(2)$ in QM

In a QM text I am using (Sakurai 2nd edition 'Modern Quantum Mechanics'), he describes two rotation groups, namely the $\mathrm{SO}(3)$ rotation group and $\mathrm{SU}(2)$ rotation group (unitary ...
2
votes
1answer
304 views

Infinite dimensional representations of $\text{SO}(3)$

In the theory of angular momentum, we wish to study the projective representations of the rotation group $\text{SO}(3)$, for which we turn to the representation theory of the double cover $\text{SU}(2)...
1
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0answers
99 views

Why do we study the projective representations of SO(3) in the context of the theory of angular momentum? [duplicate]

I have been studying the group theoretic formalism of quantum mechanics and I have yet to find a satisfying explanation for the need for projective representations in the theory of angular momentum. I ...
0
votes
1answer
560 views

Do Galilean boosts and Lorentz boosts share the same generators?

Gottfried and Yan's Quantum Mechanics introduces a generator $N$, called the boost, which generates Galileo transformations. I think in other terminology one might say $N$ generates Galilean boosts, ...
1
vote
1answer
553 views

Generalized Pauli matrix for spin larger than 1/2 [duplicate]

for Spin 1/2, we have Pauli matrix as in wiki. So what's the generalized 3-by-3 Pauli matrix for spin 1 or even larger spin? Is there a generalization method?
3
votes
3answers
647 views

Why can the Clebsch-Gordan coefficients always be chosen real?

I am talking about the angular momentum. Is there any deep reason? For some other group, it is not the case?
20
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6answers
4k views

Why is there this relationship between quaternions and Pauli matrices?

I've just started studying quantum mechanics, and I've come across this correlation between Pauli matrices ($\sigma_i$) and quaternions which I can't grasp: namely, that $i\sigma_1$, $i\sigma_2$ and $...
5
votes
2answers
1k views

Is there a relation between spin and the spin group?

In Quantum Mechanics spin appears as one type of angular momentum. Indeed, in Quantum Mechanics one angular momentum on the state space $\mathcal{E}$ is a triplet of observables $\mathbf{J}=(J_1,J_2,...
3
votes
1answer
467 views

Tensor product representation of $SO(3)$ in the Hilbert space of particle with spin $S$

For a particle with a spin $S$, the rotation operator is given by $$ e^{iJ_i\theta/\hbar} $$ where $J_i$ is the component of the total angular momentum along the direction of the rotation axis. The ...
3
votes
1answer
135 views

Using symmetry to determine a hydrogen electron's decay route from $|300\rangle$ to $|100\rangle$

Lets say we have an electron in state $|nlm\rangle = |300\rangle$ of the hydrogen atom. By selection rules, we know that it can only decay to ground state in 3 ways, namely through the $|21m\rangle$ ...
5
votes
1answer
867 views

Why is there no 1/3 spin? [duplicate]

Why do no particles have a 1/3 spin? Why are all particles' spin either a half-integer or integer? How would a particle with such a spin behave, as a fermion, boson, or neither?
1
vote
1answer
408 views

Matrix represenation of total angular momentum operator

I see that for total ket in QM of hydrogen atom we define a tensor product of kets of spatial and spin spaces, upon which spatial and spin operators, operate respectively. For the total angular ...
2
votes
1answer
328 views

3D isotropic oscillator and angular momentum algebra

In our QM class, the prof said: "We are ready to begin constructing the individual states of the 3D isotropic harmonic oscillator system. The key property is that the states must organize ...