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## Quantum Mechanics: total angular momentum of an electron in a hydrogen atom

An electron in a hydrogen atom occupies the combined position and spin state.

$$\Psi$$$$\left(\vec{r},\xi\right)$$=$$\left(\sqrt{1/3}Y^{1}_{0}\xi_{+}+\sqrt{2/3}Y^{1}_{1}\xi_{-}\right)$$

What are the possible measured values of $$J^{2}$$ (where J is the total angular momentum of the electron L + S) and with what probability will each be found?

$$J^{2}$$ = $$\hbar^{2}j\left(j+1\right)$$

$$\left|l-s\right|\geq j \geq l+s$$ , where $$l$$ and $$s$$ are the orbital angular momentum and spin angular momentum quantum numbers, respectively.

I know that, according to the given position state of the electron and the fact that it is an electron, $$l = 1$$ and $$s = 1/2$$.

I know that $$j$$ will be $$1/2$$ or $$3/2$$. Therefore, $$J^{2}$$, when measured, will be either $$3/4\hbar^{2}$$ or $$15/4\hbar^{2}$$.

I am having trouble determining the associated probabilities of the possible measurement values for $$J^{2}$$. From what I have read about addition of angular momentum, it would seem that I would need to calculate the Clebsch-Gordon coefficients for the total angular momentum. I am not really sure where to start. I have read Griffiths' explanation for angular momentum and the Clebsch-Gordon coefficients but he doesn't explain how to use them for total angular momentum.

I feel dumb asking this kind of question but I am having trouble understanding Quantum Mechanics.
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