# Worked Problems in Physics

## Nielsen and Chuang Exercise 2.58

Posted in Nielsen/Chuang by rpmuller on March 16, 2010

Exercise 2.58 Supposed we prepare a quantum system in an eigenstate ${\left|\psi\right>}$ of some observable ${M}$, with corresponding eigenvalue ${m}$. What is the average observed value of ${M}$, and the standard deviation.

As long as the state does not decohere, the average value will always be exactly ${\left<\psi|M|\psi\right>=m}$, and the standard deviation will be 0. This does not violate the uncertainty principle, which is about pairs of noncommuting observables.

## Nielsen and Chuang Exercise 2.57

Posted in Nielsen/Chuang by rpmuller on March 16, 2010

Exercise 2.57 Cascaded measurements are single measurements. Suppose ${\{L_l\}}$ and ${\{M_m\}}$ are two sets of measurement operators. Show that a measurement defined by the measurement operators ${\{L_l\}}$ followed by a measurement defined by the measurement operators ${\{M_m\}}$ is physically equivalent to a single measurement defined by the measurement operators ${\{N_{lm}\}}$ with the representation ${N_{lm}=M_mL_m}$.

Suppose we consider the matrix element ${\left<\mu|M_mL_l|\nu\right>}$ of the specified cascaded measurement. We can insert a complete set of states between the two operators via ${\sum_\lambda\left<\mu|M_m|\lambda\right>\left<\lambda|L_l|\nu\right>}$, making this equivalent to the matrix multiplication of elements of ${N_{lm}=M_mL_l}$.