## Activity: Working with Representations on the Ring

The following are 2 different represtations for the $\textbf{same}$ state on a quantum ring $$\left|{\Phi}\right\rangle = \sqrt\frac{1}{2}\left|{2}\right\rangle -\sqrt\frac{ 1}{4}\left|{0}\right\rangle +i\sqrt\frac{ 1}{4}\left|{-2}\right\rangle$$ $$\Phi(\phi) \doteq \sqrt {\frac{1}{8 \pi r_0}} \left( \sqrt{2}e^{i 2 \phi} -1 + ie^{-i 2 \phi} \right)$$

1. Write down the matrix representation for the same state.

2. With all 3 representations, calculate the probability that a measurement of $L_z$ will yield $0\hbar$, $-2\hbar$, $2\hbar$.

3. If you measured the $z$-component of angular momentum to be $2\hbar$, what would the state of the particle be immediately after the measurement is made?

4. What is the probability that a measurement of energy, $E$, will yield $0\frac{\hbar^2}{I}$?,$2\frac{\hbar^2}{I}$?,$4\frac{\hbar^2}{I}$?

5. If you measured the energy of the state to be $2\frac{\hbar^2}{I}$, what would the state of the particle be immediately after the measurement is made?

Author Information
Dustin Treece and Corinne Manogue
Learning Outcomes