Consider two states on the Hilbert space \(|uangle\) and \(|vangle\). Construct the operator \(\mathbb{U}_{u v}\) that maps one to another:

\[\begin{equation*}|vangle=\mathbb{U}_{u v}|uangle \tag{3.158}\end{equation*}\]

Express \(\mathbb{U}_{u v}\) as the outer product of Dirac bras and kets. Is the operator \(\mathbb{U}_{u v}\) unitary? Can you make it unitary?