Consider the following operators that correspond to exponentiating the momentum and position operators, \(\hat{x}\) and \(\hat{p}\) :

\[\begin{equation*}\hat{X}=e^{i \sqrt{\frac{2 \pi c}{\hbar}} \hat{x}}, \quad \hat{P}=e^{i \sqrt{\frac{2 \pi}{c \hbar}} \hat{p}} \text {. } \tag{4.153}\end{equation*}\]

Here, \(c\) is some constant.

(a) What are the units of the constant \(c\) ? From that, can you provide an interpretation of \(c\) ?

(b) Calculate the commutator of \(\hat{X}\) and \(\hat{P},[\hat{X}, \hat{P}]\). What does this mean?

(c) What is the range of eigenvalues of \(\hat{x}\) and \(\hat{p}\) for which the operators \(\hat{X}\) and \(\hat{P}\) are single-valued? Remember, \(e^{i \pi}=e^{i 3 \pi}\), for example.

(d) Determine the eigenstates and eigenvalues of the exponentiated momentum operator, \(\hat{P}\), in the position basis. That is, what values can \(\lambda\) take and what function \(f_{\lambda}(x)\) satisfies

\[\begin{equation*}\hat{P} f_{\lambda}(x)=\lambda f_{\lambda}(x) ? \tag{4.154}\end{equation*}\]

Do the eigenvalues \(\lambda\) have to be real valued? Why or why not?

(e) Using part (b), what can you say about the eigenstates of \(\hat{X}\) ?

(f) What is the uncertainty principle for operators \(\hat{X}\) and \(\hat{P}\) ? Can you reconcile that with the Heisenberg uncertainty principle?

Think about the consequences of the answer to part (c).