An instanton is a quantum mechanical excitation that is localized in space like a particle. They are closely related to solitary waves or solitons that were first observed in the mid-nineteenth century as a traveling wave in a canal. \({ }^{12}\) We'll study a model for instantons in this problem. Our simple model will be the following. We will consider a quantum system constrained on a circle, and we can define states on this circle by their winding number \(n\), the number of times that the instanton wraps around the circle before it connects back to itself (think about winding a string around a cylinder and then tying it back together after \(n\) times around). The winding number \(n\) can be any integer, positive, negative, or zero, and the sign of the winding number encodes the direction in which it is wrapped.

States with different \(n\) are orthogonal, so we will consider the Hilbert space as spanned by the set of states \(\{|nangle\}_{n=-\infty}^{\infty}\) which are orthonormal:

\[\begin{equation*}\langle m \mid nangle=\delta_{m n}, \tag{4.157}\end{equation*}\]

and we will assume they are complete.

(a) On this Hilbert space, we can define a hopping operator \(\hat{\mathcal{O}}\) which is defined to act on the basis elements as

\[\begin{equation*}\hat{\mathcal{O}}|nangle=|n+1angle . \tag{4.158}\end{equation*}\]

Show that this means that \(\hat{\mathcal{O}}\) is unitary.

(b) Assume that the state \(|\psiangle\) is an eigenstate of \(\hat{\mathcal{O}}\) with eigenvalue defined by an angle \(\theta\) :

\[\begin{equation*}\hat{\mathcal{O}}|\psiangle=e^{i \theta}|\psiangle . \tag{4.159}\end{equation*}\]

Express the state \(|\psiangle\) as a linear combination of the winding states \(|nangle\).

(c) The Hamiltonian for this winding system \(\hat{H}\) is defined to act as

\[\begin{equation*}\hat{H}|nangle=|n| E_{0}|nangle \tag{4.160}\end{equation*}\]

where \(E_{0}\) is a fixed energy and \(|n|\) is the absolute value of the winding number \(n\). Calculate the commutator of the hopping operator and the Hamiltonian, \([\hat{H}, \hat{\mathcal{O}}]\).

(d) Determine the time dependence of the state \(|\psiangle\); that is, evaluate

\[\begin{equation*}|\psi(t)angle=e^{-i \frac{\hat{t} t}{\hbar}}|\psiangle \tag{4.161}\end{equation*}\]