Chapter 12.3, Problem 12.8P

(a)

To determine

To show $\rho =\frac{1}{2}\left(1+\text{a}\cdot \sigma \right)$.

(b)

To determine

To show $\rho$ represents a pure state for $\left|a\right|=1$, and $\left|a\right|<1$ for mixed state.

(c)

To determine

The probability of measurement of $S_z$ at the North Pole, center of the sphere, and the South Pole.

(d)

To determine

The spinor $X$ denoting the pure state of the system.