3. Rotation of spin states. [10 points] We define the operator Rn(a), with a real and n a unit vector, by inSn Rn(a) = exp h = exp where we noted that S. = =n . . (a) Using the definition of the exponential function and properties of the o-matrices show that Ra(a) = I cos - 2 - io . n sin Verify by direct computation that R,(a) is unitary. (b) For brevity we write Ry(a) for Re, (a) Evaluate the operator Ry(a ) S, Ry(a)t in terms of ST, Sy, and S.. (c) Find the state obtained by acting with Ry(a) on (+). For what operator is the resulting state an eigenstate with eigenvalue //2. Explain why we can think of Ry (a) as a rotation operator. (Similarly, one can show that Re(a) is a rotation operator around an axis pointing along n.)