(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have two spin-1/2 particles and I need to calculate their Clebsch-Gordan coefficients.

2. Relevant equations

3. The attempt at a solution

I followed the procedure of applying [itex]J_- [/itex] to [itex]|{j,m}\rangle [/itex]

and [itex]J_{1-}[/itex] and [itex]J_{2-}[/itex] to [itex]|{m_1,m_2}\rangle [/itex] and comparing them. I got correctly

[itex]\langle{1,1}|{1/2,1/2}\rangle =1 [/itex],

[itex]\langle{1,0}|{1/2,-1/2}\rangle=1/\sqrt{2} [/itex],

[itex]\langle{1,0}|{-1/2,1/2}\rangle =1/\sqrt{2} [/itex],

[itex]\langle{1,-1}|{-1/2,-1/2}\rangle=1 [/itex].

Now I want to find [itex]\langle{0,0}|{1/2,-1/2}\rangle [/itex]

and [itex]\langle{0,0}|{-1/2,1/2}\rangle [/itex].

Therefore I denoted [itex]|{0,0}\rangle = \alpha|{1/2,-1/2}\rangle + \beta|{-1/2,1/2}\rangle [/itex]

and used the normalization condition [itex]|\alpha|^2 + |\beta|^2 = 1[/itex] and orthogonality to the [itex]|{1,0}\rangle[/itex] state. I got the equation

[itex]|\alpha|^2 = 1/2[/itex] from which there are 2 options:

[itex]\alpha = 1/\sqrt{2}[/itex] and [itex]\alpha = -1/\sqrt{2}[/itex] (only real coefficients by convention). How do I know which is the right option out of the two?

Thanks!

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# Homework Help: Calculation of Clebsch-Gordan coefficients

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