Central Forces 2022
Consider the following normalized quantum state on a unit ring (\(r_0 = 1\)):
\begin{equation}
\Phi(\phi)=\sqrt\frac{8}{3 \pi } \sin^{2}\left( 3\,\phi \right)\cos \left( \phi \right)
\end{equation}

Translate this state into eigenfunction, bra/ket, and matrix representations. Remember that you can use any of these representations in the following calculations.

If you measured the \(z\)-component of angular momentum, what is the
probability that you would obtain \(\hbar\)? \(-3\hbar\)? \(-7\hbar\)?

If you measured the \(z\)-component of angular momentum, what other possible
values could you obtain with non-zero probability?

If you measured the energy, what is the
probability that you would obtain \(\frac{\hbar^2}{2 I}\)? \(\frac{4\hbar^2 }{ 2
I}\)? \(\frac{25\hbar^2 }{ 2 I}\)?

If you measured the energy, what other possible values could you obtain with
non-zero probability?

What is the probability that the particle can be found in the region \(0<\phi<
\frac{\pi}{4}\)? In the region \(\frac{\pi}{4}<\phi< \frac{3 \pi}{4}\)?

Plot this wave function.

What is the expectation value of \(L_z\) in this state?