2pi rotation of angular momentum eigenket

[blue_leaf77]

Homework Statement

Prove that $e^{2\pi i \mathbf{n\cdot J}/\hbar} |j,m\rangle = (-1)^{2j}|j,m\rangle$. This equation is from Ballentine's QM book. The term in front of the ket state in the LHS is a rotation operator through $2\pi$ angle about an arbitrary direction $\mathbf{n}$.

Homework Equations

Above

The Attempt at a Solution

I can prove this for spin one half particle using the identity $(\mathbf{ \sigma \cdot n})^2 = 1$, but not for an arbitrary j. Does he simply quote this from the result of E. P. Wigner's work, as also stated in the book?

 

[TSny]

In my edition of Ballentine (1st edition), he outlines the proof in the remaining part of the paragraph where the equation is given. I think I follow his argument. See if you can pinpoint where you have difficulty with his reasoning.

 

[blue_leaf77]

On a second thought it makes sense if I visualize it as a vector in R³ rotated about arbitrary direction by $2\pi$, it should go back to its original position. But it becomes a somewhat delicate issue for half-integer spin states. Maybe I should go through the entire section first and see if it proves also for half-integer spins.