This activity lets students explore translating a wavefunction that isn't obviously made up of eigenstates at first glance into ket and matrix form. Then students explore wave functions, probabilities in a region, expectation values, and what wavefunctions can tell you about measurements of \(L_z\).
Consider the following normalized quantum state on a unit ring: \begin{equation} \Phi(\phi)=\sqrt\frac{8}{3 \pi r_0} \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.
- What is the expectation value of \(L_z\) in this state?
The wave function and it's probability density are plotted below. (I have set \(r_0=1\) to make the plotting easier). What features of these graphs (if any) tell you the expectation value of \(L_z\) in this state?
- What is the probability that the particle can be found in the region \(0<\phi< \frac{\pi}{4}\)? Repeat your calculation in the region \(\frac{\pi}{4}<\phi< \frac{3 \pi}{4}\)?
Probability v. Probability Density: Students struggle with the two different ways of finding probability: for discrete and continuous measurements. Most recognize that they need to do an integral for a continuous quantity, but are not sure when to square (before integration or after). \begin{align} \left|\int \phi_n^*(x)\Psi(x,t) dx\right|^2 \, \, \, \, \, \, vs. \, \, \, \, \, \, \int\left|\Psi(t) \right|^2 dx \end{align}
In particular, many students will forget to do the squaring for the calculation on the left because \(\int \phi_n^*(x) \Psi(x,t) dx\) looks a lot like \(\int \Psi^*(x,t) \Psi(x,t) dx\).