All Questions
Tagged with group-theory rotation
82
questions
2
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98
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$Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$
This question is inspired by my recent question How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$? with answer https://...
- 14.4k
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4
answers
408
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?
Disclaimer: I'm sure this has been asked 100 times before, but I can't find the question asked or answered quite like this. If there are specific duplicates that could give me a simple satisfactory ...
- 14.4k
0
votes
1
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168
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Can you ever obtain a pure rotation from composing Lorentz transformations?
An exercise asks one to show that given $v, u$ speeds much smaller than $c$ and oriented orthagonally, the composition of the lorentz boosts $B(\mathbf{v})B(\mathbf{u})B(\mathbf{-v})B(\mathbf{-u})$ is ...
- 75
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Rotation and translation of a function of a 3D vector
I want to change the frame by doing translation and rotation.
$$f(\vec{v})=\sum_{n,l,m}R_{nl}(v)Y_{lm}(\hat{v})f_{nlm}^v.$$
Let, $\mathcal{R}$ be the rotation matrix and $\mathcal{T}$ be the ...
- 11
3
votes
1
answer
132
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Rotation of spherical harmonics
I have a question about the rotation of spherical harmonics. In Wikipedia it is mentioned that if we make a rotation in 3D space: $R\vec{r}=\vec{r}'$,then the Spherical Harmonics can be written as a ...
0
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1
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Rotation of Pauli Vectors with $SU(2)$ reproduces the $SO(3)$ matrix. but do all $SU(2)$ matrices reproduces $SO(3)$?
So we can write the $SU(2)$ matrices multiplication as this.
$$\begin{bmatrix}\alpha&\beta\\-\beta^*&\alpha^*\end{bmatrix}\begin{bmatrix}z&x-iy\\x+iy&-z\end{bmatrix}\begin{bmatrix}\...
- 15
0
votes
2
answers
103
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Representation of groups
Can someone please explain to me the last sentence (in bold) from the following excerpt? It's from a set of lecture notes on classical fields and GR (Ch.2 Groups and representations, p.16). I went ...
- 373
2
votes
2
answers
228
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Topological proof of spin-statistics theorem confusion
I am currently studying the spin-statistics theorem. I have found a section on John Baez's website which presents a "proof" of the spin-statistics theorem. He states the theorem as:
This is ...
- 1,261
1
vote
0
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65
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What is the meaning of the parameters in the axis-angle exponential representation of $\rm SO(3)$?
In the axis-angle parametrization, an element of the rotation group $\rm SO(3)$ is written as
$$R_{\hat{n}}(\theta)=\exp\left[-i(\vec{J}\cdot\hat{n})\theta\right]$$
where $\theta$ represents the angle ...
- 11.8k
0
votes
2
answers
238
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What is meant by "rotation group"?
What do physicists mean by the term "rotation group"? Is it synonymous with $SO(3)$? Is it synonymous with $SU(2)$?
I am confused because rotations in real 3D Euclidean space can also be ...
- 11.8k
0
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2
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76
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What is the derivative of general 3D rotation with respect to one angular component? [closed]
For a general rotation $R(t_1, t_2, t_3)$ where the $t_i$'s are the components of the rotation vector in the axis-angle representation. Is there closed formula for the derivative of $dR/dt_i$?
I only ...
- 111
0
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2
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72
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Computing rotation matrices - non-commuting operators
A rotation matrix parametrized by Euler ZYZ angles, $\alpha, \beta, \gamma$ can be written as:
$$
\hat{R}(\alpha, \beta, \gamma) =
\exp{\left( -i\alpha\hat{J}_{z} \right)} \cdot
\exp{\left( -i\beta\...
- 11
0
votes
1
answer
242
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Generators of Angular Momentum
This semester I'm taking Quantum Mechanics II and we are in the theory of angular momentum. One particular thing got me thinking: when one does the representation of the rotations of the $SO(3)$, a ...
- 346
2
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0
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272
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What does it mean an electron needs 720 degrees to return to original state? [duplicate]
I have read that the electron spin is represented by a vector of 2 complex numbers. And a frequently asked question is how can it be that an electron must be rotated 720 degrees to return to its ...
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1
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121
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How do I combine the unitary rotation operators about the $x$, $y$ and $z$ axes to get the unitary rotation operator about a generic axis $u$? [closed]
I have the following in my lecture notes
In a past evaluation I was asked to combine the rotation operators about the $x$, $y$ and $z$ axes to get the rotation operator about a generic axis $u$ with $...
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