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Update after @M. Enns's answer Consider a circular orbit of radius $a$ passing through the centre of a central force is given by the equation $r=2a\cos\theta$. Then using the orbit equation one can sh …

What is the reason that $SU(2)$ group enters quantum mechanics in the context of rotation but not $SO(3)$? What really rotates and which space it rotates? It cannot be the physical electron that rotat …

You must remember that $\textbf{r}$ is an operator and to compute $\nabla\cdot\hat r$ you must act it on a function of coordinates. Here is how I derived it.\begin{equation} \textbf{L}^2=(\textbf{r}\t …

The components of the classical angular momentum $L_i$, satisfy the Poisson bracket relation $$\{L_i,L_j\}=\epsilon_{ijk}L_k,\tag{1}$$ and forms a Lie algebra (i.e, anti-symmetric, obeys the Jacobi i …

In classical mechanics, the orbital angular momentum of a particle is defined as $\textbf{L}=\textbf{r}\times\textbf{p}$. This is zero in the rest frame of the particle where $\textbf{p}=0$. Quantum …

It can be shown that the total angular momentum of a free electromagnetic field is given by (for example, in the book A Modern Introduction to quantum field theory by Maggiore, page 98, Eq. (4.82)) $$ …

Consider the Heisenberg model where the Hamiltonian $$H= J\sum_{\langle i,j\rangle}\textbf{s}_i\cdot \textbf{s}_j$$ has continuous rotational symmetry. Since $\textbf{s}_i\in\mathbb{R}^3$, the rotatio …

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 …

In the book A Introductory Course Of Particle Physics by Palash B. Pal, the author claims the following (on page 167, section 6.4.3) For single-particle systems, intrinsic parity is the only contr …

In the quantization of free electromagnetic field, it is found that the left-circularly polarised photons corrsponds to helicity $\vec{S}\cdot\hat p=+\hbar$ and right-circularly polarised photons cor …

In Quantum Mechanics (QM), angular momentum turn out to be the generator of rotational symmetry. This is trivial to see because in QM, angular momenta are defined by the commutation relations $$[J_j,J …

The conserved charge associated with the Lorentz transformation of a scalar field is given by $Q^{\alpha\beta}=\int d^3x\frac{1}{2}(x^\alpha T^{0\beta}-x^\beta T^{0\alpha})$. The quantities $Q^{ij}$ a …

Angular momenta are the generators of rotations. The orbital angular momentum, denoted by $\textbf{L}=(L_1,L_2,L_3)$, generates rotations in three-dimensional space in the planes $xy$, $yz$, $zx$ resp …

We know, that for SU(2) representations $$\textbf{2}\otimes \textbf{2}=\textbf{3}\oplus \textbf{1}$$ where $\textbf{2}$ stands for the fundamental representation of SU(2). This means that we have a s …

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 t …