## QM Ring Function

• Central Forces 2022 Consider the following normalized quantum state on a unit ring ($r_0 = 1$): $$\Phi(\phi)=\sqrt\frac{8}{3 \pi } \sin^{2}\left( 3\,\phi \right)\cos \left( \phi \right)$$
1. Translate this state into eigenfunction, bra/ket, and matrix representations. Remember that you can use any of these representations in the following calculations.
2. If you measured the $z$-component of angular momentum, what is the probability that you would obtain $\hbar$? $-3\hbar$? $-7\hbar$?
3. If you measured the $z$-component of angular momentum, what other possible values could you obtain with non-zero probability?
4. If you measured the energy, what is the probability that you would obtain $\frac{\hbar^2}{2 I}$? $\frac{4\hbar^2 }{ 2 I}$? $\frac{25\hbar^2 }{ 2 I}$?
5. If you measured the energy, what other possible values could you obtain with non-zero probability?
6. What is the probability that the particle can be found in the region $0<\phi< \frac{\pi}{4}$? In the region $\frac{\pi}{4}<\phi< \frac{3 \pi}{4}$?
7. Plot this wave function.
8. What is the expectation value of $L_z$ in this state?