Chapter 12.3, Problem 12.6P

(a)

To determine

Proof of the properties ${\rho }^{†}=\rho$, $\text{Tr}\left(\rho \right)=1$, $\langle A\rangle =\text{Tr}\left(\rho \text{A}\right)$, and $i\hslash \frac{d\widehat{p}}{dt}=\left[\widehat{H},\widehat{p}\right]$.

(b)

To determine

To show that $\text{Tr}\left({\rho }^2\right)\le 1$, and equal to $1$ only if $\rho$ represents a pure state.

(c)

To determine

To show that ${\rho }^2=\rho$ if and only if $\rho$ represents a pure state.