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 Oct 10 comment Reference for the polar parameterization of quaternions @Trimok: Perfect, thank you! Jul 2 comment How to evaluate this sum of coupling coefficients? Specifically: $$n \in \left[0,\infty\right)$$ $$l,l_1,l_2,\lambda_1,\lambda_2 \in \left[0,n\right]$$ $$m \in \left[-l,l\right]$$ $$m_1 \in \left[-l_1,l_1\right]$$ $$m_2 \in \left[-l_2,l_2\right]$$ $$\mu_1 \in \left[-\lambda_1,\lambda_1\right]$$ $$\mu_2 \in \left[-\lambda_2,\lambda_2\right]$$ ($n$ is an even integer, all other indices are integers) Jun 26 comment How to evaluate this sum of coupling coefficients? @Vibert: $n \geq 0$ is an even integer, $0 \leq l \leq n$ is an integer, all other indices take integer values and their limits follow from the definition of the CG coefficients and 6j symbols. Sorry about not stating that before. Mar 14 comment General procedure for Clebsch-Gordan expansions Thank you,I need to find the analogous equation that you gave for the spherical harmonics above, for harmonic functions on $S^3$; do you know of a book/paper where I can find the derivation you mentioned? Jan 24 comment Coupling Coefficients in SO(4) I have noticed one thing that might help, the author of eq. 1 states that "The canonical basis states of the symmetric (class-one) irreducible representation $l=l_{(n)}$ for the chain $SO(n)\supset SO(n-1)\supset...\supset SO(3)\supset SO(2)$ are labelled by the $(n-2)$-tuple $M=(l_{(n-1)},N)=(l_{(n-1)},...,l_{(3)},m_{(2)})$ of integers $l_{(n)}\geq l_{(n-1)}\geq ...\geq l_{(3)}\geq |m_{(2)}|$". Also in his notation here I believe "SO(n) irreducible representation $l_{(n)}=l$ [has] SO(n-1) irrep labels $l_{(n-1)}=l'$". Jan 21 comment What is an isoscalar factor? Its in eqs. 4.6-4.7 from "Coupling coefficients of SO(n)and integrals involving Jacobi and Gegenbauer polynomials" by Sigitas Alisauskas, you can find it at iopscience.iop.org/0305-4470/35/34/307 Jun 29 comment Clebsch-Gordan Identity @Dan: The sums are over all valid values of the arguments, specifically $-a\leq\alpha\leq a, -b\leq\beta\leq b, -d\leq\delta\leq d$ May 26 comment Why are two eigen-state-kets with different eigenvalues orthogonal? yes, I was making this mistake. Thank you for clarifying, I think part of the confusion was the idea that the indices can take non-positive values (i.e. for $j_1=1$, $m_1$ can take any of $\{-1,0,1\}$ so $\vec{e}_m$ can be any of $\{ \vec{e}_{-1}, \vec{e}_0, \vec{e}_1 \}$. All of which are mutually orthogonal. Am I understanding this correctly now? May 26 comment Why are two eigen-state-kets with different eigenvalues orthogonal? @Peter: so I assume the proof would be something like: define $\left|j_1 j_2 m_1 m_2\right> = \sum_j \sum_m \left \left|j_1 j_2 j m\right>$, where $\left$ are the Clebsch-Gordan coefficients. Do the same thing for a different ket, say, $\left|j_1' j_2 m_1' m_2\right>$, then their inner-product is the sum of the products of their coefficients, which should be zero (I assume there will be some kronecker deltas in there somewhere too). May 26 comment Why are two eigen-state-kets with different eigenvalues orthogonal? @Peter: thank you for your explanation that was very helpful. So if I'm understanding you correctly the dual bra to a given eigenket (in my case a simultaneous eigenket) is defined to give 0 as their inner-product. If this is correct, then in practical terms, given some numerical values for $j_1, j_2, m_1, m_2, j, m$, how do I compute the inner product? (i.e. the definition we've imposed tells me what the inner-product of two orthogonal kets is, but it tells me nothing about what the inner-product of two general kets is so that I can check if they are orthogonal). May 26 comment Why are two eigen-state-kets with different eigenvalues orthogonal? I admit I'm a beginner and probably am making A LOT of beginner's mistakes. I know the derivation is incorrect, I want to know what I did wrong so that I can correctly understand. May 24 comment state vector notation @David: thank you, your explanation was very illuminating. Perhaps I will chat about this a bit later, it would definitely be beneficial for me. I think armed with your explanation I'm going to try to read some more from Sakurai's Modern Quantum Mechanics. May 24 comment state vector notation @Lubos Motl: So is this to say that in a manner analogous to the similarity transformation $B=P^{-1}AP$ the function of the Clebsch-Gordan coefficients is that of the matrix P, i.e. if A is a matrix whose columns are the basis kets for some hilbert space H1 and B is a matrix whose columns are the basis kets for some other hilbert space H2 then the similarity transformation (change of basis) is effected by using the matrix P whose elements are the Clebsch-Gordan coefficients? May 24 comment state vector notation @dbrane: so is there any difference between $\left|jm\right>$, $\left|j,m\right>$, $\left|j;m\right>$? May 18 comment Integral of the product of three spherical harmonics @Ramashalanka: Arfken mentions the identity but doesn't provide a derivation, what I'm interested in is the actual derivation.