A closely related system to the infinite square well is a particle on a ring. The Hamiltonian for a particle on a ring has 0 potential energy and the position on the ring can be represented by an angle \(\phi \in[0,2 \pi)\). The wavefunction can then be expressed exclusively as a function of \(\phi, \psi(\phi)\), and it must be periodic: \(\psi(\phi)=\) \(\psi(\phi+2 \pi)\).

(a) If the ring has radius \(a\) and the mass of the particle is \(m\), write the Hamiltonian for this system in terms of the provided quantities. What is the momentum operator \(\hat{p}\) on the ring and is it Hermitian?

(b) Determine the energy eigenstates of the Hamiltonian for the particle on the ring. What is the smallest allowed energy on the ring?

(c) Now, consider the uncertainty principle for the position and momentum operators on the ring. From our derivation in Chap. 4, we would expect that this uncertainty principle is

\[\begin{equation*}\sigma_{\phi} \sigma_{p} \geq \frac{\hbar}{2} \tag{5.76}\end{equation*}\]

Is this satisfied for every energy eigenstate on the ring? If not, can you identify what in our derivation of the uncertainty principle in Chap. 4 fails or does not apply for the particle on the ring?