- #1
Kara386
- 208
- 2
Homework Statement
The question asks me to calculate the non-zero Clebsch Gordan coefficients
##\langle j_1+j_2, j_s-2|j_1,m_1;j_2,m_2\rangle##
Where ##j_s=j_1+j_2##.
Homework Equations
The Attempt at a Solution
The ##j_s-2## part is the ##m## of a ##|j,m\rangle## and I know that m has to equal ##m_1+m_2## or that equation would just be zero. And I also know that I should apply the lowering operators ##J_-## to one of these two equations:
##\langle j_s, j_s-1|j_1,j_1-1;j_2,j_2\rangle = \sqrt{\frac{j_1}{j_s}}##
Or
##\langle j_s, j_s-1|j_1,j_1;j_2,j_2-1\rangle = \sqrt{\frac{j_2}{j_s}}##
I don't know which of these two I should be applying the lowering operator to. And then I don't know how to apply it because there are two ##j, m## sets in the ket, so when calculating the coefficient using ##\sqrt{j+m}\sqrt{j-m+1}## which m and j exactly is this thing supposed to change? How do you apply the operator to an inner product? And you can't apply ##J_-## to the RHS so I'm not sure what to do about that either.
Essentially we're being taught via lots of examples, but it's hard to extract the rules from the examples without an explanation. Which is why my questions probably involve incredibly basic and fundamental concepts. Basically I have no idea what's going on, but it isn't for lack of reading or trying. I've definitely looked in lots of textbooks and on the internet, I just don't understand what they say! So any help is hugely appreciated, I'm trying to use this question to work out all the stuff I don't know. :)