How do I find this state |j,m=j> to calculate another state?

[valanna]

I’m confused about how you find the vector |s;s⟩ to use in the general equation
|θ,ϕ⟩=exp(−iϕS₃) * exp(−iθS₂) |s;s⟩
For spin Coherent states (From http://www.scholarpedia.org/article/Coherent_state_(Quantum_mechanics)#4._Spin_Coherent_States
Eq 12)
Or
how you find the vector |j,m=j⟩ to use in the equation
|θ,ϕ⟩=exp(iθ[J_x*sin(ϕ)−J_y*cos(ϕ)]) |j,m=j⟩
(From https://arxiv.org/pdf/0805.1264v1.pdf
Eq 14)

For the above state |j,m=j⟩ in the paper it appears to be assumed you should just know how to find this. I know that it is an eigenstate but I don’t know how to go from there to get that vector so that I can solve for |θ,ϕ⟩
I need it for j=4 but I’d like to be able to understand how to get it for any j and understand why?

 

[A. Neumaier]

As explained in your first link, $|s,s\rangle$ is the normalized eigenvector of $S_3$ to the eigenvalue $s$. So you pick the representation you have, write down the operator $S_3$ in a basis of this representation, and find the eigenvector numerically.

 

[valanna]

Thank you,
Sorry I missed that, I've figured out what I need now thank you so much