Problem 12.8 (a) Show that the most general density matrix for a spin-1/2 particle can be written in terms of three real numbers (a1, a2, a3): P = (1 + a3) (a) +ia2) (a-ia₂)) (1-a3) = +a σ), (12.38) where 01, 02, 03 are the three Pauli matrices. Hint: It has to be hermitian, and its trace must be 1. (b) In the literature, a is known as the Bloch vector. Show that p represents a pure state if and only if |a| == 1, and for a mixed state |a] < 1. Hint: Use Problem 12.6(c). Thus every density matrix for a spin-1/2 particle corresponds to a point in the Bloch sphere, of radius 1. Points on the surface are pure states, and points inside are mixed states. (c) What is the probability that a measurement of S, would return the value +h/2, if the tip of the Bloch vector is at (i) the north pole (a = (0, 0, 1)), (ii) the center of the sphere (a (0, 0, 0)), (iii) the south pole (a = (0, 0, -1))? = (d) Find the spinor x representing the (pure) state of the system, if the Bloch vector lies on the equator, at azimuthal angle .