Quantum Mechanics: Addition of 3 spin 1/2 particles

I had trouble finding a solution to this online, so figured I'd try making a video of it! I hope it makes some sense :). Your eigenvectors for mixed states may look different than mine depending on what parameters you set for your dependent variables!

[TimeMachine]

Hello sir, first of all: great video! Just a couple days ago, I was calculating this problem myself with the method of angular momentum ladder operators and Clebsch-Gordan coefficients and I wasn't entirely sure about my result, so I consulted Google, and here we are. I'm no expert on the topic, and I might be 3 years late, but I actually think you missed 2 states in your final result:

So your states no 1, 2, 5, 8 are all correct, I have these too. (although at no 2 it should say 3/2 instead of 1/2, minor error).
Now your states 3 and 4, and 6 and 7 are also correct, however I think because we are dealing with indistinguishable particles, no 4 is just a permutation of no 3, and no 7 is just a permutation of no 6. We should end up with a quadruplet (s=3/2) and two doublets (with s=1/2 each), but you only calculated one of the doublets, so out of the total 8 states, you essentially only found 6.

If my calculations are correct, the two missing states should look something like this:

|1/2, 1/2 > = 1/sqrt(6) ( 2 |uud> - |udu> - |duu>)
|1/2, -1/2 > = 1/sqrt(6) ( |udd> + |dud> - 2 |ddu>)

In case you read this and feel motivated, might be worth to double check, or if I'm wrong, tell me why :D Either way, appreciate the video! thumbs up

[DietterichLabs]

This is great. Nice job filling a void!

[Richardqm]

Liked the video, i was kinda losst on this :).
But the down down down in the matrix should be 15/4, couse u had (-1/2 )* (-1/2). And if u sum m1+m2+m3 on the last one u get -3/2, what only would be possible if u got s= 3/2

[dfk]

What's that app u're using in the beginning?