In Example 6.1, we introduced the Hermitian number and phase operators \(\hat{N}\) and \(\hat{\Theta}\) constructed from the raising and lowering operators, \(\hat{a}^{\dagger}\) and \(\hat{a}\). In this exercise, we will study more properties of those operators and construct their commutation relation. Recall that the relationship with the raising and lowering operators is\[\begin{equation*}\hat{a}^{\dagger}=\sqrt{\hat{N}} e^{-i \hat{\Theta}}, \quad \hat{a}=e^{i \hat{\Theta}} \sqrt{\hat{N}} . \tag{6.128}\end{equation*}\]

5 An extensive review of supersymmetric quantum mechanics can be found in F. Cooper, A. Khare, and U. Sukhatme, "Supersymmetry and quantum mechanics," Phys. Rep. 251, 267-385 (1995).

(a) Express an eigenstate \(|nangle\) of the number operator \(\hat{N}\) in terms of the continuous eigenstates \(|\thetaangle\) of the phase operator \(\hat{\Theta}\) on \(\theta \in[0,2 \pi)\).
Hint: Write the state \(|nangle\) as a continuous sum over phase eigenstates like

\[\begin{equation*}|nangle=\int_{0}^{2 \pi} d \theta c_{\theta}|\thetaangle \tag{6.129}\end{equation*}\]
for some complex coefficients \(c_{\theta}\).
(b) What is the commutation relation of the number and phase operators?
Nota Bene: Remember that \(n=0\) subtlety from Example 6.1? What is the corresponding uncertainty principle for the number and phase operators?