# Coupled angular momentum

1. Apr 8, 2007

### UrbanXrisis

1. The problem statement, all variables and given/known data

say the coupled angular momentum eigenstate for two p elections is $$|2211>$$. I am asked to operate on the eigenstate with $$L^2$$ and $$L_{z}$$ to verify $$lm$$

2. Relevant equations

$$L^2 = L_1^2 +L_2^2 +2L_{1z}L_{2z}+(L_{1+}L_{2-}+L_{1-}L_{2+})$$

$$L_{+} |l,m>=|l,m+1>$$
$$L_{-} |l,m>=|l,m-1>$$

Eigenvalues:
$$L^2=\hbar^2 l(l+1)$$
$$L_z=\hbar m$$

3. The attempt at a solution

Using:

$$L^2 = L_1^2 +L_2^2 +2L_{1z}L_{2z}+(L_{1+}L_{2-}+L_{1-}L_{2+})|2211>$$

I got it down to:

$$2 \hbar |11>|11>+2 \hbar |11>|11>+2 \hbar |11>|11>+|12>|10>+|10>|12>$$

I guess my question is how to calculate |10>|12> and |12>|10> and if |10>|12>=|12>|10>=0 and why?

Also, I was wondering why $$|2211>=C_{11}|11>|11>$$ and not $$|2211>=C_{02}|10>|12>+C_{20}|12>|10>+C_{12}|11>|11>$$

thanks,
ux

Last edited: Apr 8, 2007