(a) A group $G$ of transformations acts on a quantum system. Briefly explain why the Born rule implies that these transformations may be represented by operators $U(g): \mathcal{H} \rightarrow \mathcal{H}$ obeying

$$\begin{aligned} U(g)^{\dagger} U(g) &=1_{\mathcal{H}} \\ U\left(g_{1}\right) U\left(g_{2}\right) &=e^{i \phi\left(g_{1}, g_{2}\right)} U\left(g_{1} \cdot g_{2}\right) \end{aligned}$$

for all $g_{1}, g_{2} \in G$, where $\phi\left(g_{1}, g_{2}\right) \in \mathbb{R}$.

What additional property does $U(g)$ have when $G$ is a group of symmetries of the Hamiltonian? Show that symmetries correspond to conserved quantities.

(b) The Coulomb Hamiltonian describing the gross structure of the hydrogen atom is invariant under time reversal, $t \mapsto-t$. Suppose we try to represent time reversal by a unitary operator $T$ obeying $U(t) T=T U(-t)$, where $U(t)$ is the time-evolution operator. Show that this would imply that hydrogen has no stable ground state.

An operator $A: \mathcal{H} \rightarrow \mathcal{H}$ is antilinear if

$$A(a|\alpha\rangle+b|\beta\rangle)=\bar{a} A|\alpha\rangle+\bar{b} A|\beta\rangle$$

for all $|\alpha\rangle,|\beta\rangle \in \mathcal{H}$ and all $a, b \in \mathbb{C}$, and antiunitary if, in addition,

$$\left\langle\beta^{\prime} \mid \alpha^{\prime}\right\rangle=\overline{\langle\beta \mid \alpha\rangle},$$

where $\left|\alpha^{\prime}\right\rangle=A|\alpha\rangle$ and $\left|\beta^{\prime}\right\rangle=A|\beta\rangle$. Show that if time reversal is instead represented by an antiunitary operator then the above instability of hydrogen is avoided.