I'm a little confused on the notation my professor used for the following integral.

\begin{equation} \int \bar{Y}_{l_f}^{m_f} \left( \dfrac{-Y_1^1 + Y_1^{-1}}{\sqrt{2}}, \dfrac{iY_1^1 + iY_1^{-1}}{\sqrt{2}}, Y_1^0 \right) Y_{l_i}^{m_i} d\Omega \end{equation} Here, the Y's are spherical harmonics using the quantum mechanical convention. The bar denotes the complex conjugate. My confusion here is with the vector notation. I know I need to split up the components and integrate them individually, but I'm unsure how to properly do that.

As an example, I'm looking for someone to evaluate $Y_{l_i}^{m_i} = Y_1^0 = \sqrt{\frac{3}{4\pi}}\cos\theta$ and $Y_{l_f}^{m_f} = Y_0^0 = \sqrt{\frac{1}{4\pi}}$, where $Y_{1}^{-1} = \sqrt{\frac{3}{8\pi}} \sin\theta e^{-i\phi}$ and $Y_{1}^{1} = -\sqrt{\frac{3}{8\pi}} \sin\theta e^{-i\phi}$ so I can understand the process. I have to evaluate 10 different initial and final states for my problem, the spherical harmonics listed are the easiest of the ones that need to be evaluated. Here, $d\Omega$ is the solid angle in the form $d\Omega = \sin\theta d\theta d\phi$.

Note: I'm not looking for a full integral solution. I just need help setting up the integrals, so a 'more simplified' integral is sufficient for me.

  • 1
    $\begingroup$ These integrals are just the matrix elements of the three spin-1 operators, between states of spins $\ell_f$ and $\ell_i$. For example, I am pretty sure all your examples have $\ell_f-\ell_i \in \{-1,0,1\}$ because for other cases the answer will be zero. The 3-j symbols express such integrals in terms of Clebsch-Gordan coefficients. But the task assigned is apparently to find the values of several specific of these 3-j symbols, so you will just have to do the integrals, making lots of use of orthogonality properties of the spherical harmonics. $\endgroup$ – Mark Fischler Apr 22 at 20:46

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