in the J.J. Sakurai’s book, the formalism of finding the matrix representation of rotation operator is general, but quite long and detail. A general treatment is necessary for understanding the topic, but i think, who will use arbitrary rotation? so, here i give a simple and direct calculation on and , for use-ready.
recall that the formalism:
since is diagonal matrix, thus
so, we have to find out the for and .
i am still trying to obtain the equation, but…..
anyway, using program can solve it without headache. ( but typing Latex is )here are some result.