1. (25 pt) The wavefunction for a particle in a ring is given by $\psi = \frac{1}{\sqrt{2\pi}}\exp(i m\phi)$. If $m = 30$, calculate the following expectation values.
(a) $<\phi>$
(b) $<\phi^2>$
(c) $<\hat{l}_z>$, where $\hat{l}_z = \frac{\hbar}{i}\frac{d}{d\phi}$
(d) $<\hat{l}_z^2>$
(e) Determine the uncertainty of $\phi$ ($<\phi^2> - <\phi>^2$) and $\hat{l}_z$ ($<\hat{l}_z^2> - <\hat{l}_z>^2$).
