A quantum system has energy eigenstates $|n\rangle$ with eigenvalues $E_{n}=n \hbar, n \in$ $\{1,2,3, \ldots\}$. An observable $Q$ is such that $Q|n\rangle=q_{n}|n\rangle$.

(a) What is the commutator of $Q$ with the Hamiltonian $H$ ?

(b) Given $q_{n}=\frac{1}{n}$, consider the state

$$|\psi\rangle \propto \sum_{n=1}^{N} \sqrt{n}|n\rangle$$

Determine:

(i) The probability of measuring $Q$ to be $1 / N$.

(ii) The probability of measuring energy $\hbar$ followed by another immediate measurement of energy $2 \hbar$.

(iii) The average of many separate measurements of $Q$, each measurement being on a state $|\psi\rangle$, as $N \rightarrow \infty$.

(c) Given $q_{1}=1$ and $q_{n}=-1$ for $n>1$, consider the state

$$|\psi\rangle \propto \sum_{n=1}^{\infty} \alpha^{n / 2}|n\rangle,$$

where $0<\alpha<1$.

(i) Show that the probability of measuring an eigenvalue $q=-1$ of $|\psi\rangle$ is

$$A+B \alpha$$

where $A$ and $B$ are integers that you should find.

(ii) Show that $\langle Q\rangle_{\psi}$ is $C+D \alpha$, where $C$ and $D$ are integers that you should find.

(iii) Given that $Q$ is measured to be $-1$ at time $t=0$, write down the state after a time $t$ has passed. What is then the subsequent probability at time $t$ of measuring the energy to be $2 \hbar$ ?