Say, I have two spin-1/2 particles in the states characterized by ##(n=2, l=1, m_l=1, m_s=1/2)##and##(n=2, l=1, m_l=1, m_s=-1/2)##. Now, using something like the jj coupling scheme, I first find out the (orbital+spin)angular momentum for the individual particles:(adsbygoogle = window.adsbygoogle || []).push({});

(i) ##| 11\rangle |\frac{1}{2}-\frac{1}{2}\rangle =\sqrt{1/3}| \frac{3}{2}\frac{1}{2}\rangle+\sqrt{2/3}|\frac{1}{2}\frac{1}{2}\rangle##

(ii)##|11\rangle|\frac{1}{2}\frac{1}{2}\rangle=|\frac{3}{2}\frac{3}{2}\rangle##

How do i proceed to find the total angular momentum of the system?

I've tried to add like this:

##\Big(| 11\rangle |\frac{1}{2}-\frac{1}{2}\rangle\Big)\Big(|11\rangle|\frac{1}{2}\frac{1}{2}\rangle\Big)= \sqrt{1/6}|3,2\rangle+\Big(\sqrt{2/3}-\sqrt{1/6}\Big)|2,2\rangle##

But, the sum of the square of the coefficients don't add up to 1! So, where did i go wrong?

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# Finding the Total Angular Momentum

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