A vector under rotation has the following property: [^Ji,^Vj]=iℏϵijk^Vk. Furthermore the projection lemma is defined as the following:
⟨k′;j,m′j|→V|k;j,mj⟩=⟨k′;j,m′j|(→V⋅→J)→J|k;j,mj⟩ℏ2j(j+1).
On page 24 of this note it is
claimed that
⟨k′;j,m′j|Sz|k;j,mj⟩=ℏmj⟨k′;j,m′j|(→S⋅→J)|k;j,mj⟩ℏ2j(j+1).
I have been starring at this for hours, and it is not obvious where the ℏm is coming from, given that it is Jz|j,mj⟩=ℏmj|j,mj⟩.
Secondly, shouldn't (→S⋅→J)=SzJz, since we are only analyzing the Sz part and not the full →S vector? I feel I may be misinterpreting notation, so any help is appreciated.