I am trying to improve my understanding of the Clebsch-Gordan coefficients. I am looking at page 5 of the following document https://courses.physics.illinois.edu/phys570/fa2013/chapter3.pdf
I have derived the result for the I = 3/2 quadruplet but am having a problem with deriving the following I = 1/2 doublet.
|1/2,1/2> = (√2/3)uud - √(1/3)√(1/2)(ud + du)u
|1/2,-1/2> = (√2/3)√(1/2)(ud + du)d - (√2/3)ddu
The Attempt at a Solution
For starters I think the coefficient (√2/3) should be √(2/3) to get the RHS. I know these states can be obtained via orthogonality, but the paper suggest they can be found using the ladder operators and CG coefficients directly. Any help would be appreciated.