[Physics] Angular Momentum Expectation Values in Spherical Coordinates

homework-and-exercisesquantum mechanics

I have a homework problem that asks:

Using the spherical harmonics calculate $\langle J_x \rangle$, $\langle J_y \rangle$, $\langle J_z \rangle$ in the state $|l,m\rangle$. Use the derivative forms of the $J_{i}$ in spherical coordinates.

I'm not looking for a direct answer, but I am trying to understand how to approach this problem. I have the representations of the $J_i$ in spherical coordinates, but how do I use these to calculate the requested expectations values?

I've asked for help for a specific part of this derivation in Mathematics stackexchange at: https://math.stackexchange.com/questions/1265825/integrating-associated-legendre-polynomials

Best Answer

You have to compute $\int d\Omega\ Y_{lm}^*(\theta,\phi)\hat{J}_iY_{lm}(\theta,\phi)$ where $d\Omega=\sin\theta d\theta d\phi$, $J_i$ are the angular momenta operators represented in position space and $Y_{lm}$ are the wavefunctions for the state $|lm\rangle$, i.e. spherical harmonics.

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