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I'm reading this unsigned powerpoint presentation of the Nilsson model in nuclear structure physics. On p. 15, they have this:

$$\langle \ell'm'_\ell|Y_{20}|\ell m_\ell\rangle = i^{\ell-\ell'}\sqrt{\frac{5}{4\pi}}\sqrt{\frac{2\ell+1}{2\ell'+1}}\langle \ell 2 m_\ell 0|\ell 2\ell' m'_\ell\rangle \langle \ell 2 0 0|\ell 2\ell' 0\rangle$$

The context, if I'm understanding correctly, is that we have states in a spherically symmetric 3-dimensional harmonic oscillator potential, and we separate them into something like $\Psi(r)\Phi_{\ell m_\ell}(\theta,\phi)$. Then we're calculating the matrix element of the $Y_{20}$ spherical harmonic between two such states. On the right-hand side, what are the symbols with 8 quantum numbers in them, and how does one compute them? Are they Clebsch-Gordan coefficients, written in some unusual way, or Wigner something-or-other coefficients? For CG or Wigner 3-j, I would expect 6 quantum numbers, not 8. The powerpoint does have a table of CG/3j symbols on the last page, which suggests to me that they are to be used in computing these things.

Also, is there a typo in this equation? It seems strange to me that the third quantum number in each set is sometimes an $\ell$

$$ |\ell 2\ell' m'_\ell\rangle, |\ell 2\ell' 0\rangle $$

but in one case is an $m_\ell$:

$$ \langle \ell 2 m_\ell 0| $$

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  • $\begingroup$ My guess is that they are the Nilsson basis states $|N,\ell,m_\ell,m_s\rangle$ defined on page 14, with $m_\ell$ in some cases its maximal value. But I don’t see how to derive that equation. $\endgroup$ – G. Smith May 26 at 3:29
  • $\begingroup$ Is not adviced MathJax not to be used in titles ? $\endgroup$ – Poutnik May 26 at 3:30
  • $\begingroup$ @Poutnik In other sites there is such a thing. Not here. $\endgroup$ – Emilio Pisanty May 26 at 9:56
  • $\begingroup$ @G.Smith This is essentially the triple-product integral for spherical harmonics. This previous thread asks how to derive it. $\endgroup$ – Emilio Pisanty May 26 at 10:49
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Those look like vanilla Clebsch-Gordan coefficients to me. The coeficient $$ \langle \ell 2 m_\ell 0|\ell 2\ell' m'_\ell\rangle $$ is the coupling coefficient from $\ell$ and 2 to form $\ell'$. The only funky thing going on is that the total angular momenta of the two subsystems is reported inside the right-hand ket. I reckon that this is fair enough - after all, those are still "good quantum numbers", i.e. each side is labelled by its eigenvalues, on the left with respect to the $L_1$, $L_2$, $L_{1z}$, $L_{2z}$ CSCO, and on the right with respect to $L_1$, $L_2$, $J$, $J_{z}$.

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