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Spinors are vectors in the representation vector space, not matrices in the image of the representation map. A Dirac spinor or bispinor transforms in the (only) irreducible representation of the Clifford algebra $\mathrm{Cl}(1,3)$. This representation is four-dimensional. A Weyl spinor transforms in an irreducible complex representation of the Lorentz ...

2

In the defining 3-dimensional representation of the 3D rotation group $SO(3)$ a $2\pi$-rotation is the identity element $$R(2\pi)~=~\mathbb{1}_{3\times 3}.\tag{1}$$ In fact, the identity element of a group is represented by the unit matrix for any group representation. For $2\times 2$-matrices, we calculate using various properties of the Pauli matrices that ...

1

Angular momentum operator $J$ is the generator for rotations on a wavefunction. That is if you have a state $|\psi\rangle$ and you want to express it in terms of coordinates that are rotated about an axis $\hat n$ by an angle $\theta$, this is given by $$\exp\left(-i\frac{J\cdot \hat n \theta}{\hbar}\right)|\psi\rangle$$ Now invariance of $|\psi\rangle$ ...

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In the specific case of the 2-dimensional representation, the coefficients are 1 so it doesn't matter much. On the other hand, for the higher-dimensional reps of $SU(2)$, the coefficients in front are not trivial, v.g. your raising operator $$X\to \sqrt{2}\left(\begin{array}{ccc}0&1&0\\0&0&1\\ 0&0&0\end{array}\right)$$ and for even ...

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First, hypothise $E=-\frac{e^2}{2a_{0}}\frac{1}{\nu^2}$ where $\nu$ is an unknown parameter. This is plausible since $-\frac{e^2}{2a_{0}}$ are simply constants that give the correct units. Then, we introduce the Runge-Lenz vector \mathbf{R}=\frac{1}{2m}\left(\mathbf{p}\times\mathbf{L}-\mathbf{L}\times\mathbf{p}\right)-e^2\frac{\mathbf{r}}{r},\...

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Of course not, in general, as the anticommutator is in the universal enveloping algebra: it is not even in the Lie algebra augmented by the identity, as evident in the specific example below. For the spin 1 representation of the algebra, $J^a_{~~bc}=-i\epsilon_{abc}$, consisting of hermitean, imaginary, antisymmetric 3×3 matrices, i.e. the adjoint ...

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