Coupled Angular Momentum sates and probability

[Ed Quanta]

Two p electrons are in the coupled angular momentum states |lml1l2>=|2,-2,11>. What is the joint probability of finding the two electrons with L1z and L2z?

Here is my thinking,

With m1 + m2 =-2, the expansion becomes

|2,-2,11>= C0-2|1,0>1|1,-2>2 + C-20|1,-2>1|1,0>2 + C-1-1|1,-1>1|1,-1>2

Now I believe I am supposed to apply the L- operator to both sides since L-|2,-2,11>=0 and since L-=L1- + L2- and we apply this to the othner side of the equatio.

However what we get does not look very pretty.

Am I on the right track? And what should I be doing to get the right answer?

 

[kanato]

Remember |m| <= l, so we a state like l=1, m=-2 does not exist.. I think you should just have the last term in your expansion.. (someone correct me if I'm wrong, because it's been a while since I've done this.

 

[Ed Quanta]

Thanks, you are totally right. I remembered l>=m but forgot that -m where m>l cannot exist. Then wouldn't it just be a 100 perent possibility that -h is the angular moment for L1 and L2?

 

[dextercioby]

I didn't really undertstand much thing of your notation...It would be perfect,if were able to use the latex...
The theorem of Clebsch & Gordan states that
$$|j,m\rangle =\sum_{j_{1},j_{2},m_{1},m_{2}} \langle j_{1},m_{1},j_{2},m_{2}|j,m\rangle |j_{1},m_{1},j_{2},m_{2}\rangle$$

,where i hope you're familiar with the notation...

Daniel.