Construct States from Clebsch-Gordon Coefficients

[DeldotB]

Homework Statement

Hello all,
Im asked to construct the state $| \frac{5}{2} , \frac{3}{2} \rangle$ from the eigenfunctions $| L, L_z\rangle$ and the electron states $| \uparrow \rangle$ and $| \downarrow \rangle$.

Homework Equations

Clebsch Gordon Coefficient's table

The Attempt at a Solution

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To be honest, I am not sure how to get started. My book does not explain how to construct states from the eigenfunctions $| L, L_z\rangle$ and the electron states $| \uparrow \rangle$ and $| \downarrow \rangle$. Can anyone point me in the right direction? Thanks!

 

[blue_leaf77]

Are you asked to write an eigenstate of the total angular momentum $\mathbf{J} = \mathbf{L} + \mathbf{S}$ (which is in this case $|5/2,3/2\rangle$) in terms of the individual eigenstates of $\mathbf{L}$ and $\mathbf{S}$? If yes, the easiest way is just to look up in the table of Clebsch-Gordan coefficients which you can find online.