Setting $\hbar=1$, the raising and lowering operators $J_{\pm}=J_{1} \pm i J_{2}$ for angular momentum satisfy

$$\left[J_{3}, J_{\pm}\right]=\pm J_{\pm}, \quad J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle,$$

where $J_{3}|j m\rangle=m|j m\rangle$. Find the matrix representation $S_{\pm}$for $J_{\pm}$in the basis

Suppose that the angular momentum of the state $\mathbf{v}=|1 m\rangle$ is measured in the direction $\mathbf{n}=(0, \sin \theta, \cos \theta)$ to be $+1$. Find the components of $\mathbf{v}$, expressing each component by a single term consisting of products of powers of $\sin (\theta / 2)$ and $\cos (\theta / 2)$ multiplied by constants.

Suppose that two measurements of a total angular momentum 1 system are made. The first is made in the third direction with value $+1$, and the second measurement is subsequently immediately made in direction $\mathbf{n}$. What is the probability that the second measurement is also $+1$ ?