5. (25 points total plus 10 points extra credit) Consider the operator N = (X +...

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5. (25 points total plus 10 points extra credit) Consider the operator N = (X + Y + Z)/√√3, where X, Y, and Z are the usual Pauli operators. (a) (5 points) Is N Hermitean? Is N unitary? Explain. (b) (10 points) Show that N2 = I. = (c) (10 points) Let R := e-iN¹/³. Write R as a simplified 2 × 2 matrix. [Hint: cos(-π/3) = 1/2 and sin(-/3) = -√3/2.] (d) (10 points extra credit) Describe the rotation of the Bloch sphere corresponding to R. [Hint: Where does R map [0) (the positive z-axis)? Where does R map |+) (the positive x-axis)? How about the positve y-axis (0) + i|1))/√2?] 5. (25 points total plus 10 points extra credit) Consider the operator N = (X + Y + Z)/√√3, where X, Y, and Z are the usual Pauli operators. (a) (5 points) Is N Hermitean? Is N unitary? Explain. (b) (10 points) Show that N2 = I. = (c) (10 points) Let R := e-iN¹/³. Write R as a simplified 2 × 2 matrix. [Hint: cos(-π/3) = 1/2 and sin(-/3) = -√3/2.] (d) (10 points extra credit) Describe the rotation of the Bloch sphere corresponding to R. [Hint: Where does R map [0) (the positive z-axis)? Where does R map |+) (the positive x-axis)? How about the positve y-axis (0) + i|1))/√2?]