Clock and shift operators. Consider an N-dimensional Hilbert space, with basis $|n\rangle$ with $n = 0, 1, 2 ... N$. Consider operators $\hat{T}$ and $\hat{U}$, which act on this N-state system by
$\hat{T}|n\rangle = |n+1\rangle$; $\hat{U}|n\rangle = e^{\frac{i2\pi n}{N}}|n\rangle$
In the definition of $\hat{T}$, the label on the ket should be understood as its value modulo N, so
$|n+N\rangle = |n\rangle$, $|n+2N\rangle = |n\rangle$, etc., like a clock.
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a) Find the matrix representations of $\hat{T}$ and $\hat{U}$ in the basis {$|n\rangle$}.
b) Show that
$\hat{U}\hat{T} = e^{\frac{i2\pi}{N}}\hat{T}\hat{U}$.
c) From the definition of adjoint, how does $\hat{T}^\dagger$ act, i.e.
$\hat{T}^\dagger|n\rangle = ?$
d) Show that the 'clock operator' $\hat{T}$ commutes with its adjoint and that it can hence be
diagonalized by a unitary basis rotation.
e) Find the eigenvalues and eigenvectors of $\hat{T}$. [Hint: consider states of the form
$|\phi\rangle = e^{i\phi n}|n\rangle$
