In the question I have L = 1 and s = 1/2. First I had to find the quantum state for the highest m = m$_l$ + m$_s$ value which I did. To find the quantum states for the next highest m value I used the lowering operator of j, as can be seen in the picture below. I understand that there has to be another state for that m value, but I don't understand how they get the values for $\alpha$ and $\beta$.
They say they find it by via orthogonality but whatever I try I can't get their values. They also state that the values can be found by using a ladder operation $j_+$ on the state which is zero. I think this is equal to zero because the quantum state corresponding to j = 1/2 and m = 3/2 doesn't exist, is this correct? But also using this method I can't find the values for $\alpha$ and $\beta$.
So I have a few questions :)
1) Why is the other quantum state orthogonal to the first, and how do I use this to find the $\alpha$ and $\beta$ values?
2) Is my assumption about the state j = 1/2 and m = 3/2 correct
3) How do they find the $\alpha$ and $\beta$ values for the ladder operation method?
Thanks so much!